The the verb phrase did not stay up late is past tense (technically only the word did actually has any tense). Notice that we use the term tense to refer to a particular form (inflection) of a verb. Technically, this is different from talking about which time it refers to.
In terms of time reference, as opposed to tense, we cannot tell from this sentence whether it refers to the past, to the present or to the future. The most natural reading is that the sentence is referring to a hypothetical present:
- We always do badly in our Maths test on Monday morning. If we didn't stay up so late we wouldn't be so tired.
Here didn't stay up so late refers to staying up late in general in the same way that I don't smoke refers to time in general. We can think of it as present time, but it is really referring to "always": the past, present and future together. Here the past simple tense is used because the not-staying-up-late situation is presented as a hypothetical one. It certainly is not a current actual situation.
However, the context could give us a reading that was clearly only about the future:
- A: I don't want to go there in the morning. We'll be really tired after playing cards all night.
- B: We could go in the morning. If we didn't stay up so late we, wouldn't be so tired.
Here we B is referring to future time, not the present. Again the past simple is used not to refer to past time, but because the situation is being presented as a hypothetical one.
However, it may equally be the case that the past tense is being used purely to refer to past time:
- When I was a boy we used to go to bed at around 4 on Sunday night, and we'd always fail our Monday morning maths test because we were so tired. Occasionally though we would go through a patch of being sensible and go to bed on time. If we didn't stay up so late we wouldn't be so tired and we'd often be top of the class.
Here the use of past tense is not because of any "hypothetical situation". It does not represent "modal remoteness". It is used simply to refer to past time.
The string didn't stay up is a past simple construction. We know this because of the past tense of the auxiliary verb did. With regards to time reference, we do not know whether this is a hypothetical conditional with backshifted verb forms or normal conditional which uses tense in the same manner as any other sentence. We can only tell this from the context of the sentence. The sentence could be referring to past time, general time, present time or future time.