Usually best to have some options in hand to give them, by phone or at the airport. I looked these up on KVS, hope it's useful info - showing the seats left in each cabin, assuming in irrops they won't be looking for X availability (which presumably means you're flying in economy):

LHR-IAD
11.15 BA217
F5 J9 W9 Y9

LHR-PHL
13.15 BA067
F2 J0 W6 Y9

LHR-BWI
13.50 BA229
J3 W6 Y9

LHR-IAD (your original flight)
17.00 BA293
F4 J9 W9 Y9

LHR-PHL
17.15 BA069
J9 W6 Y9

Also loads of room on LHR-JFK-DCA throughout the day.

So basically all the flights are pretty empty in economy, hard to imagine they can't just move you across, especially as there are actually X fare seats on most of these. FYI, there's also Club and WT+ award availability on many of them.