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."
These phrases are not synonymous in English. Waking up is what you do before you get up.
They mean different things in English, as in French.
They are not the same. You can wake up and still continue to stay in bed. When you get up, you had better be awake. "I just woke up, but I think I will stay in bed for a few more minutes ." "I just woke up. I better get out of bed and get ready for work."
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