I've been here for almost 8 months and I'm just now getting off my lazy bum to start the process (no excuse, I know, but I didn't realize there was a separate provisional licence requirement and it looks like this may add a fair bit of time to things).
The provisional licence should only take about 3 weeks or so to arrive (according to the
government website), so if you applied for it now, you should have it by early June. You will need the provisional licence in order to take the theory and practical tests.
Specifically, since I don't own a car and don't know anyone here who could 'accompany' me as a valid UK driver, how would I provide an L-plated car and what would I need to do to make everything kosher? Do driving instructors provide a service for something like this?
Since you don't currently have a car, I'm assuming you haven't really driven in the UK so far? If you're not planning to drive on a daily basis, you don't really need to worry about having a L-plated car and an accompanying licenced driver for day-to-day use.
In this case, in order to pass the tests, I think your best bet would be to take some driving lessons with a UK instructor. The instructor will provide the car, the L-plates and will be your 'accompanying licenced driver' during your lessons and you will be able to take the practical test in their car - so no need to worry about getting a rental car.
Hopefully you can pass the tests before your 12 months is up, but if not, you'll just have to only drive with an instructor if you don't have a car/licenced driver to accompany you.
Will the DSA even check the entry stamp on my entry clearance?
Not sure about the DVLA, but if you are stopped by the police and are found to be driving on an expired US licence, you can get into big trouble.