You should be able to get your provisional in a week or two.
Once you have that, you can take the theory test.
Once you've passed your theory test, you can take the practical. There's no requirement to take lessons, but the do help. Even if you've driven in the US for years, the test is tough to pass.
You can book a practical quickly by snagging a cancellation. I'm not sure if anyone can do that or only instructors - my instructor booked mine that way.
I'd driven for over 10 years in the US and I still took 10-15 hours of lessons. I felt they were worthwhile. Somethings you need to know to do the UK way to pass the test and they're probably not the way you were taught in the US.
I'd suggest taking a lesson or two with an instructor, see what they think you'll need, and then decide if you want intensive lessons.
If you're in London, I can give you the name of my instructor. I didn't take intensive lessons from him, but I thought he was good. YMMV