Generally, US citizens are granted a visa at customs upon port of entry. It is discretionary though which general means that they will only not give visas if one has a significant criminal record or plainly does not have the funds to finance their stay.
Given that you were so denied last time I would advise contacting the nearest British embassy and applying for a visa before go. You should do this immediately because you do not have a lot of time. Information on the requirements (and other info) is available here:-
You should take pf of your identity and address, as well as bank statements showing the funds available to you. Call ahead of your trip and ask if other documents are required.
If you do not get it in time then you may have to take a risk and travel, or not take a risk and don't I'm afraid.
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