All great advice from the above forum members.
Totally agree...Don't stay in Stratford the night before your flight....Move closer to Heathrow, even if it's just to a town or city that has quick transportation to the airport.
If you still want to do Stratford-Upon-Avon at the end of your trip, you might leave London and go to Stratford-Upon-Avon for 2 nights, leave Stratford-Upon-Avon and take the train to Oxford (which is right down the train line) for one night. Spend the night there and then the next morning take the coach (bus) directly to Heathrow airport.
There is not a train line directly from Oxford to Heathrow, but the coach will get you there. The coaches in England are very nice....comfortable, clean seats (more comfy than a seat on a plane), WiFi, and you're riding up high so you have a great view of the countryside or whatever towns you're passing through. Coach fare is inexpensive.
Yet another option is to take the train from Stratford-Upon-Avon to Windsor (a train change is necessary mid-journey). Spend the night in Windsor. There are several good hotels there, good restaurants, cafes, pubs. You will probably arrive early enough in the day to drop your bags at your hotel and tour Windsor Castle. Windsor is a lovely town for a walk afterwards. It's right on the Thames River with boats rides available.
Windsor is very, very close to Heathrow. About a 25 minute taxi ride if traffic is not bad. You'll be delivered right to the front door of your terminal.
Take a taxi to Heathrow the next day. It's a short journey. Our taxi ride cost us 20 pounds, but that was a couple of years ago, so count on about 30-35 pounds now.
We on this forum can provide you with names and phone numbers of taxi services if you decide to go with Windsor.
Personal preference: I really love Windsor.