Another option is to route through ORD or JFK, so you fly into LHR. You would have to leave DFW a few hours earlier.

I would agree with those who say that it is pretty much of a gamble to attempt this on the roads, particularly if you will arrive on a weekday. Realize that if your bags are transported from LGW to LHR, that would be by road, they would be taking that same gamble. You can take bags yourself on the train and on the tube, have done it many times, not my favorite chore after an all-night plane ride, but definitely doable.

I've never done this transfer, don't need that kind of stress in my life. Good luck.