I have never experienced any problem what so ever with carry-ons when flying first, 777 or 744. As Scott said earlier, if it doesn't fit overhead, I'd trust the cabin crew to handle it for you. In my experience, it will magically and safely be returned to you just before or after landing.

And other than an unpleasant experience many many years ago, I've not had any baggage problems with BAA once inside the terminal.