Changing your name on your US passport via using a UK deed poll would require the info here, I think.
https://uk.usembassy.gov/u-s-citizen-services/u-s-passports/change-name-on-u-s-passport/ The deed poll to change my name in the UK basically required that I write a paragraph to the effect that I was abandoning the old name and would from that point be using the new one. I believe I also had to take it to the JP court (possibly?) there in Scotland to have it made official. That was it. I was then able to use that document to change my bank account info, utilities, etc.
[My situation was a little different from yours in that I had changed my name in the USA prior to moving to the UK so I already had a US passport in my new name, but as I was there on an Irish passport, I had to continue to use my old "Irish" name until their laws allowed me to change it on a new Irish passport. Ireland does not recognize UK deed polls, but once I could provide proof of two years' of use of the new name they issued me a revised passport. I do not remember if I notified the UK immigration service when I did the deed poll or if I waited until I had the new passport in hand.]
With the massive backlog in getting US passports processed at present, I would have to wonder what implications this would have on any visas or other immigration documents you have that are issued by the UK. I would assume that when your new US passport arrived you'd have to then send it off to the UK authorities to process with the deed poll, but don't know much about the timing - if you should wait until you have the new passport or if you should send the deed poll in ASAP (if you go that route). Hopefully someone here might know, or you could speak with an immigration attorney to be sure.
