On a slightly different tack there's the aadvantage map if you have a reasonably tight date window. You search one day and one region at a time but at least you're seeing real availability.

https://www.aa.com/awardMap/home.do

(I would search one way ex-LON initially for ideas [include partners, and non-stop to narrow it almost entirely to BA].)

Or you can just ask for ideas here. For summer in particular, with children, I don't find BA's longhaul options terribly broad.