I've always had success with V and X fares which are the 2 lowest above BE. BUT, I don't take chances with the wait list. I call the DM line and ask them to look for available dates. They've always found something within a day or two. Although I've never had to do it, I'm willing to take a nearby destination and fly there instead since Europe has so many airports close to each other by air or rail. You can also open jaw with one leg to or from your preferred destination.

For LHR I'd be willing to take DUB, AMS, BRU, or CDG, all a short flight with a LCC.

Good luck.