I woke up late does not mean you got out of bed. I woke up late: 我睡醒了比较晚。 I got up late: 我起床起得晚。

You can say :我睡过头了

Actually it depends. if you got up and some guys asked you. You can say 我起晚了。

我睡過頭了 means I overslept I would say 我很晚才起床

It‘s OK you use 我起床晚了，but 我睡过头了 always means that you want to apologize to somebody