So here I will talk about this three constructions -
Let's not wait.
Don't let's wait.
and
Let's don't wait.
The contraction form of let us in imperative use is let's. But if we say only this about let's we will miss out on many more.
Because often you will find following constructions -
- Let's us wait till Tuesday.
The use of us there is redundant. We not only find objective case after after let's, at time we find subjective cases also, like the followings -
- Let's you and I wait till Tuesday.
- Let's you and me wait till Tuesday.
These are as redundant as us. Obvious question comes when such redundancy is allowed or whether they are at all correct. As long as it's in casual informal situation, or in speech where it's seen as mistaken repetition of the pronoun that is already present in the contraction, it's okay. But overall such usage is generally criticized as redundant especially by earlier commentators. It is however to be noted that Quirk et al. 1985 characterizes let's us as "familiar American English".
The Cambridge Grammar of the English Language (CGEL) says that the use of let's you and I/me ... is so common that it qualifies as acceptable informal style in Standard English.
CGEL page 935 -

But generally the subjective case of pronoun after let is not standard, but still there are examples where we can find such usage -
- Let we [1485 lete vs] hold us together till it be day. - 1634 Malory's Arthur
We can't conclude that it's older usage. Here is a much modern example compared to the one above -
- ... a restaurant in Queen Street way we cud get good dhal pourri and chicken curry, let we go. - Samuel Selvon, A Brighter Sun, 1952
Then there are examples like this -
- Let fortune go to hell for it, not I - Shakespeare, The Merchant of Venice, 1957
- Let he who made thee answer that - Lord Byron, Cain, 1821
Jespersen says that the last sentence allowed he because of the relative clause and who. In the relative clause the subject is represented by who, and in turn it's he. But the later editors changed he to him. Notwithstanding the construction persists in Modern English. Let's check the comic-strip paraphrase of famous passage (John 8:7) of the King James Bible:
- Let he who is without sin cast the first stone - Joe Martin, "Mister Boffo", 25 Oct. 1986
Jespersen further says that the reason why there is subjective case rather than an objective case in those sentences is because the pronoun after let is perceived as the subject of the following infinitive, rather than the object of let. He calls it Notional or Virtual Subject.
The Oxford English Dictionary, however, calls such occurrence of subjective case incorrect.

By the 20th century let's is frequently treated as a unit, rather than as a contraction, and then the pronoun follows.
The negative of let's is formed in three ways -
let's not ... -> widely used
don't let's ... -> mainly in British English
let's don't ... -> chiefly in American English
Examples -
- In all events, let's don't celebrate it until it has done something - Alexander Woollcott, letter, 26 Jan. 1918
- But don't let's harp on it.
The Columbia Guide to Standard American English by Kenneth George Wilson -

The Qualls Concise English Grammar by Eduard J. Qualls -

Encyclopedic Graded Grammar Vol 2 -

The Cambridge Grammar of the English Language -

References -
- The Cambridge Grammar of English Language by Huddleston & Pullum 2002
- A Comprehensive Grammar of English Language by Quirk et al. 1985
- Merriam-Webster's Dictionary of English Usage
- Fowler's Modern English Usage
- The Columbia Guide to Standard American English by Kenneth George Wilson
- The Qualls Concise English Grammar by Eduard J. Qualls
- Encyclopedic Graded Grammar Vol 2
- The Oxford English Dictionary (OED)