What is the difference between come up and come over in the following contexts?
One of the dishes I ordered was missing, so I asked the waiter to come up/come over.
A police officer came up to/come over to me in the street.
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There are a couple of different distinctions, depending on the context you're dealing with. For starters, if you're actually describing a situation where there is some difference in height between the two locations (e.g. you are dining on the second floor and the waiter is currently downstairs), then "come up" would generally be the more appropriate phrase because you are literally talking about them moving higher in order to come towards you ("come over" is more commonly used when there is no (significant) difference in height between the two locations).
Assuming, however, that you are in a situation where everybody is at roughly the same altitude, then the phrase "come up to" has somewhat different implications than the (more neutral) "come over".
"come over" really just means that somebody is moving from a location that is further away to one which is closer. It does not say anything about how close they get, but the implication is that they "come over" close enough to do whatever needs to be done, which may or may not be that close depending on the purpose (e.g. a waiter might "come over" to stand next to your table, or a policeman might "come over" to keep an eye on a suspicious person while still staying across the street from them).
"come up to", however, strongly implies that somebody comes right up next to something/someone else before stopping. Because of this, it can also have implications of somebody getting in the way (either physically blocking somebody, or just disturbing whatever they were in the middle of doing). You would not generally ask somebody to "come up" to you, unless you actually meant that they should come and stand right in front of your face. This is, however, why people often talk about people (such as police officers) "coming up" to someone in the street; the implication being that they are actually approaching with the intent of interrupting/stopping them for some purpose, and that they actually get fairly close in order to do that.
So regarding your two example sentences, you would say:
One of the dishes I ordered was missing, so I asked the waiter to come over.
because you don't actually want the waiter to come stand in front of your face, you just want them to come close enough that you can tell them what's wrong. Whereas, it would be common for someone to say:
A police officer came up to me in the street.
if you actually mean to say that the officer approached you to ask you questions, or to stop you from going somewhere, etc (i.e. to interrupt what you were doing for some reason).
come up to someone means to approach someone: not what one would say about a waiter.
THOUGH: one could imagine:
The waiter came up to me as I was leaving the restaurant. [approached] The police officer came up to us on the esplanade. [approached]
The police officer came over to my car [from his side of the road] The waiter came over to our table [from the bar]. Can you come over to my house tomorrow [from yours]?
come over has an implied prepositional phrase. [Don't ask. :)]