Woken or Have woken

It depends what you are trying to say. I assume you mean

I woke up late [this morning] and had to take a taxi to attend the first lecture.

Woken up late suggests I was woken up late (that is, someone else woke you up).

Woken up late would be a wrong phrase when referring to yourself, doing the action of waking up yourself. It would probably mean that somebody else was doing the task of waking you up, so woken up late would be the passive form of 'She/he woke me up late', which is I was 'woken' up late(i.e.somebody else was doing this). Had woken would also not be the best choice since your second statement is in the simple past tense, so it wouldn't be quite right to have your first one in past perfect tense. Hence the best way to say this according to me would be: Woke up late. I had to take a taxi to attend the first lecture.