I did my PPL in Florida on a now defunct school, and have used my license since that all around Europe and the US. Few points:
The visa is a non issue, couple of forms, a phone call and 3-4 hours at the embassy. Remember to budget for this though.
It IS possible to get your PPL in 3 weeks, but you will not enjoy any of that time. It is very stressful. 5 weeks, weather permitting, is a realistic schedule when you plan to fly every day. Even that is tight.
Fort Pierce is a nice airport.
After you receive your EASA license, you do not have to convert anything to fly in the UK. It is the same license you would get in Europe.
BUT, the fact that you have it doesn't mean you are proficient on using it. UK has many many idiosyncrasies about its airspace and ATC. After learning the radio in the US, and opening your mouth in the UK with that knowledge, you're in for a real surprise with all the odd "basic service" and "pass your message" and freecalls. Expect an hour or two with an instructor and maybe 5 hours of self-study to get used to the UK habits. Also, you probably won't see a grass strip during your time there, whereas they seem to be very popular in the UK. Thats easy though.
I wouldn't go to Florida for the cost. It will be cheaper, but not decisively so. I would go there for the experience. Learning to fly in a busy airport in the US makes you VERY proficient on the radio, and you won't have the fear many UK pilots I know have, of flying to controlled airfields, even abroad.
Also, weather is more predictable there. Give or take the odd hurricane, summer months are: sunny morning, CB's start building up afternoon, then thunderstorms for a few hours, and a nice clear evening. As long as you are at the airport every morning at 0630, ready to fly at 0700, and if needed ready to fly again at 1700, and assuming no tech cancellations and instructor availability ok, you'll be done in 3 weeks.