I've done all my training in the US because I lived there and I find it hard to imagine that it would be any cheaper for a UK resident to go there and get a PPL. Firstly this idea that you can knock it out in 3 weeks just does not seem realistic. Yes the weather in California and Florida can be nice but they're certainly not immune from schedule busting IMC days, turbulence and convection sigmets, marine layers and winds that Orcadians would recognize. Then there is always the personnel and equipment scheduling issues that every school has no matter where you are. I suppose there as some Cracker Jack type folks that may at a stretch and with a bit luck get it done in 3 weeks but I suspect that for the 99% of us 'average Joe's' it would end up with spending a whole bunch more time and money on unplanned lessons, repeatedly extending hotel and rental car bookings, airline change fees, unexpected work absence and in the unfortunate event of a check ride failure sitting around for weeks waiting to get back on the examiners schedule while racking up several thousand more in accommodation and living expenses.
If I was starting out on a PPL and was determined to do it overseas I think I'd do it in Southern Europe, as a hop home prompted by choice or necessity is only a two hour ride and 100 quid on the discount airlines and returning back for more training would not raise any visa issues.
Again, that's just my opinion and as they say opinions are like arseholes, everybody's got one.