Hello All,
Is it OK to formally speak/ write "I am done" or "I will be done in a few minutes"..

I have heard the Americans speak like this a lot.

Is it accepted in formal UK English? If not, how would they typically say it?

I seem to remember an English teacher saying something like "A roast might be done, dear. You meant, I'm sure, that you are finished."

Regardless, I'm quite likely to say "done." I hope she's not rolling in her grave.
I'm from the UK, and in formal written English I would use "I have finished" and "I will be/have finished" instead. Both your sentences are fine in everyday British English though (although the contractions "I'm" and "I'll" would often be used).
 BarbaraPA's reply was promoted to an answer.