Hi there,
I recently had my embassy interview for my J-1 exchange visa last week. I was fortunate enough to have my appointment processed very quickly and I was in and out of the embassy within 30 minutes. However, I heard some people can take up to four hours so make sure you bring every required document! My visa was dispatched the following week and they advised me it would normally take between 5-10 working days from the time of the interview to the visa being dispatched home. (delivery time is 1-2 days through the courier)
The US embassy keep hold of your British passport whilst they process your visa application. However, if you need to fly back to Scotland, they offer the option to keep your passport and then send it to them at a later date. (This would allow you to fly home after the interview), HOWEVER, you do not have much time between your interview date and your enrollment date. Keeping your passport would only delay your application so I advise against it as time isn't on your side. I would suggest researching whether you can fly internal British flights without a passport.
I hope this helped.