The US and the UK were signatories to the Budapest Memorandum (all three memoranda, actually, there are similar ones with Belarus and Kazakhstan), but it was never intended as a mutual assistance treaty in the way the North Atlantic Treaty (the “NAT” part of “NATO”) is. It was just an agreement to respect each other’s territorial integrity and not to use weapons against each other. It literally says:
The Russian Federation, […] reaffirm their obligation to refrain from the threat or use of force against the territorial integrity or political independence of Ukraine, and that none of their weapons will ever be used against Ukraine except in self-defense or otherwise in accordance with the Charter of the United Nations.
The cop-out clause, of course, was “except in self-defence”, which is what Russia implicitly claims, when saying that its citizens in Donbas, and thus Russia itself, were under attack by Ukraine. Playing the victim has always been the preferred way to justify a war of aggression.
The part about giving up the nuclear weapons is implicit in the preamble which welcomes Ukraine to the non-proliferation treaty as a non-nuclear-weapon state.