This process can take up to 60 days.
Indeed - if you are coming from the UK - this process is, currently, endless.
There is no process avialable, at the moment, to convert a UK licence to an FAA one. You would need to take the full FAA flight test.
One POSSIBLE way MAY be to transfer your UK JAA licence into some other countries JAA licence - and then do the 60 day route.
It's a snag with UK law - not really the CAA's fault (certainly not all of it) - but there is no transfer process available.