Peedal is correct, if you have a fiance visa, then you are already approved for marriage. However, if you are getting married in England and Wales, you would have to give notice in person at one of the designated registries. I've emailed the Home Office for a list of designated registries, and my fiance has asked the York register office, where we were planning to give notice and eventually marry, if they are one of the designated offices. I don't know if that info is available yet.
Erm, a Chuch of England Wedding isn't an option for people who aren't Christian.
You would still marry at the registry in your district, so it doesn't involve disrupting your wedding plans. You just might have to take a trip elsewhere to give notice. I just hope the designated registries are distributed evenly around England and Wales.
The thing is, when we booked our wedding date at the York register office, we were told that we had to make an appointment to go there and give notice at least a month in advance of the date we are actually going to give notice-- the waiting list is that long. If they limit the number of registries where you can give notice, there are going to be awfully long queues.