I did it three years ago and didn't have too much trouble. Nobody seemed to really know what the rules were, they finally stuck me in a room by myself with a policeman / security guard that barely spoke English. But he looked over my guns, permit, and ammunition then gave me the OK to be on my way.
I'd fly into Gatwick if that's a choice, the airport is much smaller and easier to navigate.
I flew into Heathrow on a domestic from Glasgow then had to switch to the International Terminal. I couldn't find a shuttle so finally just hiked over there with my huge guncase and duffles in tow on a cart. I'd say the walk was over a mile, maybe two, and it was no fun at all toting that gear.
Destry