I have a U.S.-based Amex card and each the of several times that I've done MR transfers to BA Miles/Avios over the past year or two, it's always been instantaneous.
If you are thinking of taking advantage of the current 50% bonus offer and have a U.S. Premium Rewards Gold Card, don't forget that you have the right to borrow up to 15,000 MR points (this no longer appears as an option on the Amex MR website, but can still be done over the phone). Borrowed MR points have to be "repaid" within a calendar year through the MR points you earn during that time, otherwise you will be charged for the balance at some rate or other of some many cents per point.