You'll need to get approved by the HPC and CSP before trying to get work, employers generally wont accept anyone even for an interview unless you have registration from both bodies, so I would get that sorted first.

It sounds like you've been looking at private companies only (if you are going through agencies in London), most physio's in the UK work for the government funded National Health Service (NHS), thier jobs in England and Wales are advertised here: NHS Jobs - NHS Careers and Jobs (search under physiotherapist rather than physical therapist) and jobs in Scotland are advertised here: nhs scotland online recruitment

Its a case of applying for each job as it comes up via the websites. To be perfectly honest you would be better off getting a visa to come over here first, live here and then apply- I don't know of anyone who has got a working visa through an NHS employer, generally you would need to be based here first (with a long term visa) and apply like everyone else. The jobs that are advertised through private agencies are usually very short term (as in weeks rather than months) or for private musculoskeletal clinics....

I'm not sure on the specifics of a US citizen getting a visa for the UK so I can't really comment on how to go about getting one of those, sorry!