You are allowed to take the Theory and Hazard Perception tests as soon as your provisional licence is granted, so it's just a case of you deciding when you are ready to take it.
In order to book the practical test, you have to pass the Theory and Hazard Perception tests first. The theory test result is valid for 2 years... so you have to take the practical within 2 years of passing the theory. Depending on which test centre you book your practical test, there may be a waiting list... my test centre had a 2-month wait for booking tests, but that was about 8 years ago!
It is difficult to know how long it will take to pass the practical test - some people pass it first time around, others take a few attempts to pass (it is not an easy test to pass as you can easily fail for stupid little things). Same with lessons - some US drivers pass easily with no lessons at all and other take several weeks of lessons before they book their test (I would say that it is wise to have at least a couple of lessons in order to learn the specific requirements of the UK driving test... they can be very fussy on how you do things - like where you put your hands on the steering wheel, or how many inches from the curb you should be when reversing around a corner etc.).
As a UK citizen learning to drive for the first time at age 17 (and a very nervous driver who wasn't in a hurry to pass the tests), my timeline was the following:
early May 2000: applied for provisional licence
late May 2000: 17th birthday - finally allowed to learn to drive
June-Dec 2000: weekly driving lessons (either 1 hour or 2 hours each week)
Dec 2000: took and passed theory test (hazard perception test had not been introduced yet)
Jan 2001: booked practical test. First date available was early April 2001.
Jan-April 2001: weekly 2-hour driving lessons
April 2001: took and failed practical test. Rebooked test for early June 2001
April-June 2001: weekly lessons
June 2001: took and failed second practical test. Rebooked test for early August 2001
June-August 2001: weekly lessons
August 2001: practical test was cancelled due to examiner off sick. Rebooked for the next week
10th August 2001: took third practical test...and finally passed

- got full licence a couple of weeks later
However, my brother's timeline was:
Sept 2007: applied for provisional licence
early October 2007: 17th birthday - first lessons
October-Feb 2008: weekly and some twice-weekly lessons, took and passed theory/hazard tests (not sure of exact date)
Feb 2008: passed practical test first time around!
Because you don't know how long it will take you to pass the tests and get the full licence, it's best to allow as much time as possible to get them before your 12-month period for driving on your US licence runs out... if you haven't got your UK licence by the end of the 12 months, you are treated as a learner driver and will only be allowed to drive accompanied by someone over 21 who has held a UK licence for at least 3 years!