I transferred any BTC I had remaining on Poloniex to Bittrex a few days ago as Poloniex refused to give any guarantees about sharing out the Bitcoin Cash.
I didn't expect Bittrex to have the Bitcoin Cash / BTC market up and running so fast, but it seems it did so right after, or not long after, the first block was mined. So, for me, it was just a case of exchaning Bitcoin Cash (which had appeared in my Bittrex wallet) for BTC.
As to verification, I think I just gave them my name, address and date of birth - I can't remember giving them any documents. There's a link here, but whether it will be like that when you open an account, who knows?
Edit: I can't remember how long it took to get verified.