BA's C seats are set in such a way that they're more private than those of UA and AA and have better storage options. (UA and possibly AA have better IFE systems, though.) FWIW, if BA is in fact an option, window seats in rows 62 and 64 upstairs in the 747s are best (due to easy aisle access), followed by any other window seats upstairs in the 747s, both because of the ambiance of that cabin and because you have extra storage next to those seats. Not sure if without BA status you can book specific C seat assignments far in advance on BA, or can do so without some extra charge. If you do fly BA, you can credit the miles to your AA account.

Between AA and UA, I agree with Shawn that UA is better if you get the flat-bed seats which are installed on all international 767s and 747s. You can find out which planes fly the ORD-LHR route at the UA website. You'd be taking a chance on a 777 with UA in terms of getting an old style C seat. And as with BA, the best UA C seats are window seats in the upstairs 747 cabins, unless of course your preference is to be on the aisle. (AA doesn't fly 747s.)