As far as how long the application process takes, it depends on how you apply. If you apply online and the mail your supporting documents to the consulate, the total time is about two weeks, on average. If you apply in person, it's a same day service for straightforward application.
What state do you live in? That will determine where you apply. (The 3 visa-issuing consulates are Los Angeles, Chicago, and New York City.)
Have a look on
www.britainusa.com for all the info, supporing documents etc.
The cost for a fiance visa is $1050. It is good for 6 months, and you must enter the UK and marry before that 6 months is up. You cannot work while on a fiance visa, but once you apply for and get teh spousal visa after the marriage you will be able to work.