The US oath of allegiance requires you to "bear true faith and allegiance" to the "Constitution and laws of the United States of America". That isn't such a bad thing, and is IMO significantly better than pledging allegiance to some batty human, elected or otherwise. It depends on whether you see the individual as sovereign, or the State.
It also requires you to "absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state or sovereignty". The UK apparently don't recognise that as renouncing UK citizenship and I see it the same way. One pays taxes to both regimes, and consider that more relevant in this day and age than paying allegiance.
Once entered the UK at LHR via the visitor's line using a US passport because the european line was significantly longer. Guy just waved me through, but you could see him deflate when he realised he couldn't give me a hard time about why I was there.
Profound stuff for we SLF