The fiance visa (K1) usually takes about six months. While you await approval, your fiance is not prohibited from visiting by LAW, but Point of Entry officers have been known to turn them away. They may believe that your fiance is impatient with the process and has decided to enter on a tourists visa to stay illegally. He or she would have to provide lots of proof of intent to return (return ticket, copy of apartment lease, letter from employer stating that they're expected back at work in a fortnight, etc.) If turned away, they would have to board the next flight home.
I don't believe you have to worry about the legalities of getting married in the US and then getting a spousal visa... especially if you don't expect any problems (the fiance has enough money to support you, you don't have children with contested custody, you don't have a criminal record.)
If, for whatever reason, you are denied a UK visa, then your spouse would have to leave the US in order to start the US visa process. He would have to leave within the 90-day tourist visa. Overstaying would screw you both up royally and you could be separated for years, as he'd likely be banned from ever returning.