AA flies direct to LHR from ORD, DFW, LAX, MIA, JFK and RDU. You'd need to route through one of these cities if you don't want to use BA. Personally, I'd probably try through MIA or DFW, and maybe have a day or two layover in one of those cities to minimize the chances of weather completely screwing up your plans.
US - Have you looked at Star Alliance partner awards too, or just direct on US? That gives you a lot more flexibility.
In both cases, I think you have to be flexible. You may not get a direct routing. You may have an additional stop along the way. As long as it's part of the same ticket, this shouldn't cost you significantly more (maybe some extra taxes/fees). Additionally, you may have to do what's called an "open jaw", where you fly into one airport and fly out of another one. An example: Let's say that you're able to find Lufthansa availability from the US to Germany, no award availability from Germany to UK and United availability from UK to US. They could still get you to Germany, and then you could book a separate ticket to get to the UK.
I understand why that's not optimal. But when you have fixed dates, you have fewer choices
