How long is your unmarried partner visa for. If it is for > 6 months and you have > 6 months left on the visa (which you should, they are good for normally 2 years, right forum)? Then you should be sucessful in applying for a Certificate of Approval (COA) from the Home Office. This would allow you to get married in the UK. Now technically, if you qualify for marrying in the Church of England, you would not need to COA, which is a loophole that government is trying to shut. I don't know all of the rules for getting married in the CoE, but I assume one or both of you have to be a member.
If you are here on a visitors visa or have less than 6 months left on your visa, your best bet is to leave the country temporarialy and apply for a fiancee visa. They are a lot more straight forward then US!
