I've had good luck flying overseas with a gun, including multiple plane changes. The problem is, before you fly, you need to read the fine print of your airline's rules and regulations on firearms. I'm OK flying United, but I need to make darned sure that, while looking for the best fare, I don't switch to some regional carrier which, even if it has a cooperative arrangement with United, may have different firearms rules. That can be a pain. And you'll certainly want to insure your gun.
I put my gun on a plane in Cedar Rapids; never see it again (after two plane changes) until I arrive in Glasgow or Edinburgh. And I make sure that I have the proper US documentation (customs form) and UK documentation (Visitors Shot Gun Permit, mailed to me in advance by the shoot where we'll be staying) all taken care of. One copy of both with the gun; one copy of both with me.