tl;dr: Apply for a UK visa in the USA.
First, welcome to the USA as a citizen!
Second, as a dual citizen myself, I know your immigration history belongs to you as a person and neither of your passports. Consider if your history would be passport and not person bound then the first thing anyone deported would do is very conveniently lose their passports with the red deported stamp. This also works in a positive direction as well: you can use a visa in an expired passport with a new passport (you can -- I crossed the US border countless times like that) because the visa is for you and not your passport.
So it is for dual citizens as well. I got my NEXUS card as a Hungarian citizen / Canadian resident and could just attach my Canadian passport when I finally got one to it because my "trusted traveller" status is about me and not my passport.
All in all, you becoming a US citizen doesn't wash you clean. It does, however, significantly reduce the risk of you becoming an illegal immigrant in the UK. (Although not always -- I was pulled once into secondary on the US border despite being a Canadian resident aka. completing the long, grueling process to legally migrate to Canada, who would want to live illegally in the USA after that? Ridiculous.) Anyways, the thing is to do is apply for a visa. Link on top.