UK Immigration Law
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If your fiancee is in the UK on a 6 month visitors visa then she cannot apply in the UK for a fiancee visa. She must leave the country and apply for a fiancee visa or appply to come as a spouse after you are married. It makes little difference in the sense that the things you have to establish for UKBA are the same for either application.I.e. that the reationhip is genuine that you will live together after the marriage and can support yourselves without recourse to UK public funds etc.
As to the US you will need to check with an US immigration lawyer but I believe you can apply for a green card after you are married.
I am currently travelling so it may be a little time (6 or 8 hours) before I can respond to any supplemental questions