Student or Learner
Would I be late for a meeting or to a meeting?
'To' would not be wrong; ems did not say that it was. She said, correctly in my opinion, "In BrE, we generally refer to being late "for" something (a noun) and simply "late" if followed by a verb." (My emphasis added)
Bear in mind that there is a difference between being "late for something" - you were supposed to do it at a specific time but you actually did it after the planned time; and "late" as a simple adjective.
I was late for work = I was supposed to arrive at 9am and I arrived at (for example) 9.05am.
I was late to bed last night = I went to bed later than normal, or at a time that other people would consider to be late. Perhaps I went to bed at midnight or 1am, but normally I go to bed at 10pm.
You're late = I expected you to be here earlier.
I hate being late for school = It makes me unhappy when I arrive at school after the time I should arrive.
He stayed late at work = He stayed at work beyond his normal working hours.
I'll be home late tonight = I will arrive home after my normal time of arrival.
I'll be home very late tonight = I will be home at a time that the hearer/speaker consider late (for example 11.30pm).
Almost all the examples given by emsr2d2 apply to AmE as well. The only exception, IMO, is "I was late to bed last night."
As we said, but perhaps not clearly enough, "late for a meeting" is correct and the most frequently heard. "Late to a meeting" is also grammatically correct but not used very often.