Here’s my idea but necessitates booking things now.
Book a room at the Hilton at Paddington Station.
Book Paddington Express tickets now to garner any sort of savings.
15 minute train ride from LHR to Paddington. Walk outside to hotel, check in, drop off bags and head right back out.
Probably about 6 or 7pm by now. Walk towards the Pride of Paddington pub a block away. On the same side of the Road as the Hilton is an HCSC Bank ATM. Gets some pounds.
Cross the road, and turn left walking back towards the Hilton to the Paddington Underground Station. Get your Oyster Card there ( 5 pounds refundable) Put 5 pounds on it and take a Bakerloo line train to Bakers Street Station. Mind the Gap exit the train and then get a Jubilee line train to Westminster.
At Westminster exit the train and walk outside to Parliament Square. Houses of Parliament always look lovely at night Unfortunately the iconic tower with Big Ben is in scaffolding.
From here a couple of walking routes: People will be out so safe walks.
Cross over Westminster Bridge walking towards the London Eye or walkup Whitehall to Trafalgar Square and then over to Covent Garden. If you like Indian food Punjab is very good. It’s purple can’t miss it.
Head back to Paddington Station Underground via the tube. From Covent Garden underground it’s a Piccadilly line train to Piccadilly Circus then a Bakerloo line train to Paddington. Another 20-25 minutes
If still awake have a pint at the Pride of Paddington or Sawyers Arms.
Next morning get up early 6:30 or 7am and walk over to the Italian Gardens in Hyde Park. Enjoy a morning coffee there or walk along the path adjacent to the Serpentine past the Princess Diana Memorial Fountain and have a coffee and a nosh at the Lido Cafe. You’ll enjoy seeing the waterfowl, joggers, walkers, swimmers and dogs.
Head back to hotel by 8:30am, retrieve luggage and hop back on an Heathrow Express train to LHR and your flight home.
That’s what I’d do during a 20 hour lay over. See some iconic sites and enjoy the ambiance of London.