Julian, the method for getting the piece of paper you need actually varies from FSDO to FSDO. The flying school which Viggen goes to should be able to advise him of the local requirements. Cost also varies from FSDO to FSDO - mine wasn't free, but that was from Scotsdale FSDO in Arizona - I don't know whether Californian FSDOs will charge or not.
But basically, Viggen, find a school, contact them, and they should be able to sort out anything you need. If you're only doing dual, you won't need anything. If you want to rent, you will need a "Private Pilot License Issued on the Basis of a Foreign License" (or something like that), which as others have said, is not difficult to get.
You were also asking about conversions/hours required. This will depend on your school - most schools will just want a check-out with an instructor as long as what you're flying there is similar to what you've been flying here. Should be a formality, but make sure you get instruction on differences between flying in US and UK (mainly charts, airspace and radio at uncontrolled fields).
Have fun!
FFF. .----------