I have also taken the early morning flights from either EWR or IAD on United, landing into Heathrow between 9:30 and 10:00 p.m. the same day. Granted, we pay for an additional night's lodging, but it's great to have a good night's rest and start the next morning refreshed, showered, and with a good English breakfast that'll keep you going until about 2:00pm!
The last time I flew into London, it took me nearly 2 hours to clear Customs (on a Friday night--what was I thinking?), so, you're right, the Heathrow Express is not a definite for these late arrivals; plus, traveling alone and in my 60s, I have a few safety concerns.
What I have done in the past and continue to do is have my London hotel set up a car and driver for me -- sometimes they have them on staff, and the charge is added to the hotel bill. Does your hotel offer this service, or can they recommend a reliable source? You might email the concierge for details.
Happy trails, Ann, and all the best,