I would suggest ignoring everyone suggesting a local mom-and-pop school, here is how it is;
First of all, time left to fly in Europe with a FAA license is running out - you can still do it in the UK, but with Part-FCL approaching quickly, I would recommend against this - current derogation from the regulation extends to April 2016. Conversion from FAA->EASA is not too difficult after you have 100 hours of total time, "training as required", a couple of written tests, and a checkride.
So if you intend to use your pilot privileges in Europe, I recommend doing an EASA course. There aren't many EASA ATOs left in the US. Couple in Florida and I think one in Arizona. EASA has an up-to-date list on their website (ATO = Approved Training Organization).
You WILL need an M-1 visa for flight training. You cannot do ANY flight training towards any license/certificate as a non-citizen/permanent resident in a Part 61 school. Only Part 141 schools can issue the I-20 form, required for your M-1 application. Don't let anyone tell you otherwise - it is what it is. TSA won't check, you most likely won't get caught, but if you do, you will be violating your status, and won't be eligible to enter the US for 5-10 years.
The visa process is as follows:
You enroll on a course with a P141 school - they will take some money off you and issue an I-20 form. You then get your SEVIS approval, and get approved at the TSA foreign flight student website.
Then, you fill up the DS160 form for the visa application. It's a bit ridiculous but it only takes about an hour to fill up.
Then, you call your local embassy and take a day off work to sit through the "interview". The interview means you sit in a stinky room for 3-11 hours (some embassies have CNN, some have nothing. Belfast is better than London btw...), you end up giving your forms, answering a "so you want to be a pilot?" question, and thats that. Your passport is mailed to you in a day or two with the visa stamped to it.
Remember to have proof of funds, they will ask for it. A printed bank statement is enough. Also remember to have the postage envelope from their sponsored postage company, they only accept one kind of env...
I've been through an INS check at a flight school once, they do happen. Don't risk it without a visa. It sounds a lot worse than it is. A day to fill out the forms, a day to sit the interview.
Fingerprinting is done in the US, that only takes a few hours, no big deal. Usually the clearance to start flight training comes in a couple of days after the prints have been taken. (some places start training before the clearance comes through, up to you).
After all this, you'll start the flight training. On a good school, it is a great experience. You end up sitting the exams at the school in the US, and then doing the checkride with an examiner the school knows.
After all this - you go to the UK, and you go to a flight school that teaches you how to fly in the UK. It is a completely different world, and you will need some hours to get used to the European way of doing things, especially if you are in the UK with the weirdest ATC oddities in all of EU.