Irrespective of where you learn, I would say that a budget of £4000 is very optimistic. I have just completed my FAA PPL and I would estimate that including all my flights, accommodation etc I spent around £7500. I have now flown 90 hours. In my case it would have been less expensive to stay in the UK.
However, the USA is ideal for basic training for a few good reasons:
Good weather means an intensive course is possible, less time spent revising earlier flights because of weather delays etc.
No landing fees, which is great for circuit practice and getting the hang of those landings. I did 188 of them in the month I was there which would have cost me a fortune at my local airfield
It is good to be able to shed your daily life for a few weeks and concentrate on just one thing, but as others have said, its no holiday.
Pros for the UK
Er... weather. Flying in clear blue skies all the time makes you very wary of flying in marginal conditions in the UK.
Airspace and RT rules and regulations are very different in the USA so you will need to revise these with an instructor when you return to the UK. If you will do most of your flying in the UK then it makes sense to learn here.
Good luck whatever you decide
