You could also apply for NEXUS, which provides the same Global Entry benefits when entering the US as well as equivalent benefits when entering Canada. At $50 for 5 years it is half the price of Global Entry alone, though the cost is that the in-person interview can only be done at the subset of Global Entry locations where CBSA officers are also present (mostly at land border crossings and Canadian airports with US pre-clearance).
The reason this is relevant to your question is that, while NEXUS used to have a residence requirement for members, it is now explicitly open to Canadians and Americans no matter where they live as indicated by the CBSA web site:
To be eligible for the NEXUS, you must be a citizen or permanent
resident of Canada or the United States.
Effective June 30, 2012, Canadian and American citizens living outside
of Canada or the United States or who have recently returned to either
country and have not previously been able to meet the three-year
residency requirement are now eligible to apply.
Note that NEXUS used to be a program separate from Global Entry (NEXUS members couldn't use the Global Entry kiosks at US airports) but got turned into GE+Canada when the US opened GE to foreigners. I believe the change to eliminate the residence requirement was done to make it match what GE allowed for Americans already; it is certainly clear that the US would not allow Canadians living in random countries to use GE machines when entering the US if they denied Americans the same privilege.
So if you are a US citizen you can apply for Global Entry where ever you reside without a prior requirement for a background check from that country. I had an American co-worker living in Hong Kong who joined Global Entry while there, so I'm sure even residence in a place with privacy laws that likely preclude such checks from abroad doesn't disqualify an American. While it could be that there are certain countries of residence which would cause them to decline a GE application from a US citizen, it is pretty certain that the UK won't be one of them.