Katie - 'To get up late' and 'to wake up late' are not exact synonyms in English or in French (se lever and se réveiller) - they are interchangeable (synonymous) in use when waking up and getting out of bed occur in close temporal proximity, although it is equally correct in that circumstance to say both - 'wake up and get up' without being completely redundant or, indeed, tautologous.
Otherwise, their meanings are different - you can wake up without getting up, and you can get up long after you have woken up - from a chair for instance.
"se lever' matches the use of 'to get up' and it would be a misunderstanding to learn it as being synonymous with the English expression 'to wake up'.
Worth having a look at the definitions on wordreference (or other dictionaries) for more detail:
I do agree with the dictionary definition of these two phrases, but when speaking in a colloquial fashion they can and are translated the same. When someone tells me they woke up at 7 am, I would never say "but when did you get up?" The question would be viewed as redundant. Often the question is asked: "When did you get up?" And the response that follows is: "I woke up at 7."
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