"I'll be late for my interview if I don't leave now." "I'll be late to my interview if I don't leave now." both fine and natural. "I'll be late for an interview if I go eat first." "I'll be late to an interview if I go eat first." If you are going to use "to an" it would sound more natural to add an adjective "I'll be late to an important interview if I..." I may be "grammatically" incorrect to say you can use either, but I have definitely, as a native, heard both ways "I'll be late for" and "I'll be late to."

I am almost certain it should be 'for'. I am not a native speaker though..

late for

"For" is more commonly used but either is actually correct. The concept for each of the words is actually a bit different. If you change the context in which they are used, you will see what I mean. Driver can you please hurry? I am late for an interview. Next time I will leave earlier. I never want to be late to an interview again. The first is used in an immediate context. I am still on my way to an event that is supposed to be happening right now. The second is talking about a non specific event. However, they are used interchangeably and either way is considered correct.

I am sure it is "for".