Do you need to live in the UK a minimum amount of time to get one, and is a spousal visa sufficient documentation? Also - how much am I looking at paying?
You don't have to live in the UK a minimum amount of time to get one, but as others have said, if you do have a US licence you can only drive on it for 12 months after you move here, after which, if you haven't got a UK licence yet, you can only drive as a 'learner'. If you don't have a US licence (or any other licence) at all, then you are a 'learner' from the beginning and will need a provisional licence in order to drive.
To get a provisional licence, you will need to send off your passport to DVLA (it costs £50 for the provisional licence). Then, when you take your tests you will need to show the provisional licence (and possibly other ID too - I'm not sure).
In terms of cost, it really depends how long it takes you to be comfortable with driving and to pass the tests.
Each hour of lessons costs approximately £25-30 (although you can often get discounts if you buy blocks of lessons). Some people only need a few hours of lessons to brush up on the skills required to take the test, other people need many more hours.
The theory test costs about £30.
The practical test costs around £70. Only around 45-50% of people pass their driving test on the first attempt, so be prepared to have to take it more than once.
I learned to drive in the UK from scratch age 17 and I was a nervous driver, so it took me a long time to be comfortable on the road. I had weekly lessons (1 or 2 hours per week at £14 per hour... this was in 2000) for 15 months before I finally passed my test and I also failed the practical test twice, so in total, I think it cost me about £1,200 to learn to drive!!
However, my youngest brother turned 17 in October and passed his test on his first attempt in February, so he spent only a fraction of what I did on driving lessons and tests.