「lately」と「late」の違い

One of my students was a little late for class earlier this week and as soon as she came into the room she said "I am sorry to have kept you waiting." (this part is OK.)

She then said "I woke up lately."

What she should have said is "I am sorry to have kept you waiting. I woke up late."

Lately means not long ago or recently. E.g. Have you studied English lately? See you in class.