I'm currently living up here in NY and it took the CAA less than a week to charge me 36 quid (including the postage time!), but the FAA took about six weeks to process everything and send me back my letter. My impression from speaking to folks at my flying club is that the response time from the FAA is highly variable.
You might not run into trouble with the issue of the temporary certificate with your UK medical, but you might find wherever you are renting the aircraft from might require a FAA one because of stupid insurance policies etc. The 'rental agreement' where I fly stipulates a US medical is required. It might be worth the $100 and one hour of your time getting this done.
The whole process is straightforward, relaxed and everyone always want to talk about flying!