I'd just sit on the Piccadilly line from St Pancras, and contemplate my life choices as it drags its shaky way to Heathrow.

If not with very heavy bags, tube (or bus if wanting to avoid ups and downs, or taxi) to Paddington then Liz line. Or as above (I didn't think of the Farringdon connection, Ldnn1).

You could even Uber it the entire way, but the train is likely to be quicker.