If you can find low-tier award seats (and that's a big if), it sometimes makes sense to buy miles directly from an airline (like AA, AS, or UA) when they are running a sale, and then redeem for an award ticket.

Another option might be to take a positioning flight to a European city outside the U.K., if the fares are significantly lower from there. A good fare-search engine, like the one at matrix.itasoftware.com can be useful in searching fares from cities that are within a 500-mile radius or so from LON.