The above is correct,
AFAIK.
In practice I would strongly recommend anyone doing this to fly with an
FAA CFI/CFII sometime - even if you are just popping up in some G-reg (you do have a UK PPL, haven't you?
) and he is an unpaid passenger.
Anybody can (and must) read the FAA PTS book, but an FAA instructor will know the US way of doing things.
There is no problem with an FAA instructor to obtain the TSA registration, and I know of UK FAA instructors who do have this.
Getting TSA as a student is also no problem. The only real gotcha is that the TSA reg is for
a specific school or instructor, and if you need to change this you may need to pay the money again. I had to pay it 3 times; once because the U.S. school was not Part 141 and thus could not be in SEVIS and thus could not issue an I-20 and thus I could not get the Visa; once because the school delivered the I-20 to a UK based FAA instructor who was out of the country at the time and the document got lost; 3rd time I got lucky and did the IR in Arizona.
The only real waste of time is the day spent at the US Embassy getting the Visa.... and the Visa is valid for a particular school only!!
I have some detailed notes on the process - email me if interested.
Currently, the standalone FAA PPL/IR can be done wholly in the UK. I know of a very decent setup which can do it.