In this case what is correct?
Keep Tom and I updated. <---- nominative case "I"
Keep Tom and me updated. <---- accusative case "me"
QUICK ANSWER IN A NUTSHELL:
Both of your versions could easily be heard in the conversation of people whose status as speakers of today's standard English is clear. But unfortunately, examples similar to your #1 version with nominative "I" ("Keep Tom and I updated") are nevertheless condemned as incorrect or illiterate by many teachers and pedants and usage manuals.
That previous paragraph is a paraphrase of a paragraph on page 107 of a highly regarded textbook (which was published 2005) by Huddleston and Pullum, A Student's Introduction to English Grammar (ASITEG):
The pattern in [57.ii.b ] (% They invited Sandy and I ) is heard constantly in the conversation of people whose status as speakers of Standard English is clear, but it is nevertheless condemned as incorrect or illiterate by many usage manuals. For this reason it is not so common in print: editors will often 'correct' it. Nonetheless, examples are certainly found. Those who condemn it simply assume that the case of a pronoun in a coordination must be the same as when it stands alone. Actual usage is in conflict with this assumption.
And so, many teachers might consider that it would be safer for a student of English to use only examples similar to version #2 with accusative "me" ("Keep Tom and me updated"), because it is supposedly correct English.
But be aware that there are many native English speakers of today's standard English who do sometimes intentionally use the nominative "I" in examples similar to yours. And so, try not to, er, correct their English.
For many speakers of today's standard English, the grammar rules for when stuff are in coordination are not always the same as the rules for stuff that are not in coordination.
For a clear cut example where a speaker of today's standard English would not treat a coordination similar to a non-coordination, there's the example in the 2002 reference grammar by Huddleston and Pullum (et al.), The Cambridge Grammar of the English Language (CGEL), on page 1326,  in subsection "Special syntactic treatment of coordinates":
where [ii ] shows "that the normal rules concerning the requirement of a determiner with count singulars are sometimes relaxed in coordinations" (CGEL page 1326). There are more examples in footnote 45 on page 1326.
But unfortunately since it is easier and simpler to teach over-generalized rules in school and textbooks, that is exactly what often happens: it is common that many over-generalized and faulty "rules" are blindly taught in school and in textbooks.
When a first person pronoun is the last coordinate in a coordination, as is done in your two examples, where a non-coordinated personal pronoun would usually be in accusative case, then, both nominative and accusative case pronouns will often be found to be used by many native English speakers in today's standard English.
Since many teachers and editors blindly use the over-generalized "rule" that a personal pronoun in a coordination must always use the case that a non-coordinated personal pronoun would use, then it's often safer to follow that rule when in those environments: that is, use "Keep Tom and me updated", which is version #2 that uses accusative "me", since the pronoun would be in accusative case in the corresponding non-coordinated pronoun in "Keep me updated".
But if you hear a native English speaker who speaks fluent English using a construction similar to "Keep Tom and I updated", then don't be so quick to correct their English. That English speaker might be one of many native English speakers who happen to speak that variety (or dialect) of standard English.
Just between you and I, the answer depends on whether or not you're in school, or writing for a boss, or writing for yourself:
If you're in school, then your teachers most likely want "Keep Tom and me updated", which is version #2 that uses accusative "me".
If you have an employer, then they probably have their own style guide and editors, and a preferred style. Most likely they too will usually prefer "Keep Tom and me updated", which is version #2 that uses accusative "me".
If you are writing for yourself, then you'll use the version that you think appropriate for the prose you're writing.
THE GRAMMAR OF IT ALL:
Your examples involve a coordination where the last coordinate is a personal pronoun, and that coordination is realizing a function that would usually be realized by a non-coordinated personal pronoun in accusative case.
TOPIC: "between you and I"
First, let's look at a more commonly documented issue: "between you and me" vs "between you and I". This issue comes up so often that it is usually discussed in a usage dictionary, such as Merriam-Webster's Dictionary of English Usage (MWDEU) published 1989/1994, or the more recent Merriam-Webster's Concise Dictionary of English Usage (MWCDEU) published 2002.
(ASIDE: Just between you and I, a decent usage dictionary, such as Merriam-Webster's Concise Dictionary of English Usage (MWCDEU), will often be helpful in questions related to standard usage.)
In my copy of Merriam-Webster's Concise Dictionary of English Usage (MWCDEU), in their entry "between you and I", the concluding paragraph on page 135 is:
Conclusion: between you and I seems now to be primarily a spoken form which no amount of correction by commentators aiming to improve written English will extinguish. Neither it nor the prescribed between you and me appears very often in print. Our little current evidence almost invariably uses me. For more instances of the anomalous use of pronouns, see MYSELF; PRONOUNS; WHO, WHOM 1.
As to grammar: Usage similar to "between you and me" is part of today's standard English, while the evaluation on usage similar to "between you and I" is perhaps not so clear-cut. According to the 2002 reference grammar by Huddleston and Pullum (et al.), The Cambridge Grammar of the English Language (CGEL), they think that usage similar to "between you and I" should be considered a variety of today's standard English (CGEL page 463, an excerpt is provided further down in this post).
So, although schools will typically teach that "between you and I" is incorrect, there are many speakers that do use that form in their variety of today's standard English.
TOPIC: a coordinate nominative pronoun corresponding to a non-coordinate accusative
Now let's look at the more general case which your examples are part of: using a coordinate nominative pronoun in places that would correspond to a non-coordinate accusative.
For some related info from a highly regarded grammar source, there's the 2005 textbook by Huddleston and Pullum, A Student's Introduction to English Grammar (ASITEG), page 107:
Case in coordinations
For many speakers the above rules extend to constructions where the pronoun is coordinated, but there are also many who use special rules for coordinative constructions. Note the status markers on the following examples:
The whole coordination is subject in [i ] and object in [ii ], so in the absence of coordination we would have nominative I in [i ] (I went over there) and accusative me in [ii ] (They invited me). Construction in [i.b ] is not accepted as Standard English, though it is very common in non-standard speech. Construction [ii.b ], however, is used by many highly educated people with social prestige in the community; it should be regarded as a variant Standard English form.
Prescriptive grammar note:
- The pattern in [57.ii.b ] is heard constantly in the conversation of people whose status as speakers of Standard English is clear, but it is nevertheless condemned as incorrect or illiterate by many usage manuals. For this reason it is not so common in print: editors will often 'correct' it. Nonetheless, examples are certainly found. Those who condemn it simply assume that the case of a pronoun in a coordination must be the same as when it stands alone. Actual usage is in conflict with this assumption. (Note that we have already come across another instance of coordinated NPs differing in form from non-coordinated ones: recall the discussion of husband and wife in [15.ii.a ].)
For some more related info, there is the 2002 reference grammar by Huddleston and Pullum (et al.), The Cambridge Grammar of the English Language (CGEL), page 463:
Coordinate nominatives corresponding to non-coordinate accusatives
a. % The present was supposed to represent [Helen and I], that was the problem.
b. % Any postgrad who has any concerns about working conditions or security in shared offices is welcome to approach either [Ann Brown or I] with them.
c. % It would be an opportunity for [you and I] to spend some time together.
d. % He had intended to leave at dawn, without [you or I] knowing anything about it.
Single pronouns replacing the coordinations would have to be in accusative case: us in [i ] and [ii.b ], them in [ii.a ]. One particularly common use of this construction is in the expression between you and I, and indeed usage manuals often discuss it under that heading. It must be emphasized, however, that these nominatives are found quite generally in coordinations functioning in positions where single accusative pronouns are used. The pattern shown in [i ], with the nominative as final coordinate, is much more common than the one in [ii ], where the nominative occurs as first (or both first and final) coordinate.
There can be little doubt that the quite common use of this construction is related to the stigmatism attaching to accusatives in subject coordinations like those in : people are taught that Me and Kim will do it and Kim and me will do it are incorrect, and many generalize their avoidance of such coordinate accusatives to other functional positions. [see note #A by F.E.] The schoolteacher's strictures focus primarily on the 1st person singular pronoun (since this is where children most commonly depart from the standard variety), and in construction , with final-only nominative, I is overwhelmingly the most frequent form that is found. Compare  with the much less widely used:
i. % They've invited [the Smiths and we] to lunch.
ii. % Liz will be back next week, so I've asked Ed to return the key to [you or she].
Because these coordinate nominatives are perceived to be associated with avoidance of stigmatized accusatives in subject coordinations, they are often described as hypercorrections. This is to imply that they are 'incorrect', not established forms in the standard language. Construction [23.i ] with I as final coordinate is, however, so common in speech and used by so broad a range of speakers that it has to be recognized as a variety of Standard English, and we will reserve the term hypercorrection for examples like [23.ii ] and .
Note that the distinction between 1st person singular and the other personal pronouns is also evident in such constructions as:
Example [i ] belongs to the same style level as They like the same kind of music as we -- i.e. it is very formal. But [ii ] is not felt to be stylistically restricted in the same way; many speakers who do not themselves use examples like [23.i ] would nevertheless feel much more comfortable with [25.ii ] than with [25.i ].
[note #A: F.E. considers that that previous sentence as worded by the authors could be misleading and he personally prefers how they explained this issue in their later 2005 textbook ASITEG.]
* == ungrammatical: * This books is mine.
# == semantically or pragmatically anomalous: # We frightened the cheese.
% == grammatical in some dialect(s) only: % He hadn't many friends.
? == of questionable grammaticality: ? Sue he gave the key.
! == non-standard: ! I can't hardly hear.
H&P CGEL is the 2002 reference grammar by Huddleston and Pullum (et al.), The Cambridge Grammar of the English Language.
ASITEG is the 2005 textbook by Huddleston and Pullum, A Student's Introduction to English Grammar.
MWCDEU is the usage dictionary Merriam-Webster's Concise Dictionary of English Usage, published 2002.
Referred to info:
There is related info in the usage dictionary Merriam-Webster's Concise Dictionary of English Usage (MWCDEU), in their entry "between you and I".
The case of the personal pronoun "I/me" when used in coordination is discussed in H&P's 2005 textbook, A Student's Introduction to English Grammar, page 107.
There is related info in the 2002 H&P CGEL, section "Coordinate nominatives corresponding to non-coordinate accusatives", on page 463, and also, page 1326,  for "Special syntactic treatment of coordinates".
Here's a related answer post: Between you and (“me” or “I”)?