This is likely a 40% chance of working, but it's the best hopeL

1. Have a car service at your pickup point (have them come early and pay the wait time just in case you get done early).

2. Tell the driver there's a significant amount in it for him such as $50 cash, over and above all charges, if he hoofs it to IAD.

3. In a mad dash hitting all DC lights and no traffic, you can make it to the curbside in 35 mins.

4. Have your BP and docs. in hand, do not check bags and make a bee-line for premium security (presuming you've got access).

5. Call BA when you are enroute and let them know you're enroute. In a true overbooking this won't help, but if there's a judgment to be made, a note in your PNR may just save you having your seat given away to somebody else while you're running down the concourse.

This is about all you can do. I would also have a backup plan ready (where will you stay at IAD overnight and people to call to let them know you won't be in UK on time).