That won't work, Nial.
While it isn't technically illegal for a non-subject to show up at Heathrow with a firearm, it might as well be. On this side of the pond, there will need to be a transfering dealer between you, and the gun, before it arrives. Somedays, form 6 shows up after three weeks, or so, some other days form 6 takes four months.
My experience, the US Customs folk will be downright unfolksy when there is a firearm involved IF you don't have the little chit, signed, by one of them, stating the gun was yours prior to you going abroad. They will ask funny question about what the gun is worth (NOT what you paid for it) and how old it is, too.
I'd expect the gun would be seized by US Customs, if you got that far, Lord knows what the English do to Yanks that run afoul of UK firearms law, and, would be locked up in bonded storage, at which point you have an arbitrary number of days until said gun is destroyed. The number of days is at the discretion of the poo-bah of the perfunctories at whatever Customs fifedom you are dealing with.
I was actually dragged into a situation very similar to this by my local office of ATF (nobody at ATF can import anything, they have to use an FFL or importer) with a South African national who got some stunningly bad advice and brought his Belgian SXS and his Morgan (auto) in a container when he immigrated here.
I got the gun released. I don't know what happened to the car.
Don't try it. I do believe there was a time when one could do this, but, for civilians, anyway, that was a long time ago.
Best,
Ted