You have only to be resident in the U.K. to be eligible for the provisional license (learner's permit), so I believe your student visa would be adequate. You'd be able to start driving as soon as you receive the provisional, which should take no more than a few weeks.
Starting from scratch without a U.S. license, you'll be under exactly the same conditions as any British learner, i.e. you'll need to have a licensed driver who has held a U.K. license for at least 3 years and is at least 21 accompanying you at all times, you must display "L" plates on the front & rear of the car, etc.