Leave from London or Oxford?

Do you mean leaving from Oxford for Heathrow? That's straightforward and no need to go into London proper. There is a direct, very regular coach service (or was pre-virus), from Oxford to the airport. You'll need to include a bit of spare time in case of traffic, of course, but it's a simple journey. The service is Oxford Bus Company's "airline". They pick up in the centre and make a few stops in the Oxford suburbs heading east to the motorway. Used it many times in both directions when we lived in Oxford.

The coach might be dear for 6 people though. Instead perhaps consider a local car service that has people carriers for 6? That might be cheaper or similar price, but with the advantage of door-to-door. We used Oxford Carriage Co. sometimes. Hopefully, they're still in business after the current unpleasantness.

In the summer of 2018 (those halcyon days), I did a two-week research workshop at the Bodleian Library. Rather than pack up and relocate to London the day before my flight (I had been there before the course), I was able to celebrate with other participants and leave directly from Oxford to LHR via Coach the next morning. Super easy and comfortable. There are many departures (every 15 minutes in those days and starting very early). I actually arrived well before check in for my American Airlines flight had begun (the 11:50 flight) and had plenty of time, after security, to browse the duty-free and eat. The Oxford bus station is centrally located (Gloucester Green) and the trip is not terribly long which makes it almost as easy as riding to LHR on the Piccadilly line.

This is the link to the bus service the other posters described. Their current schedule is constrained because of Covid, but even now you could leave at 8:00 and be at Terminal 5 at 9:20, for example. I haven't ridden from Oxford to LHR but it worked very well in the other direction. You shouldn't have any trouble with a morning ride for an afternoon flight.

If you are worried about travel to Heathrow from Oxford, you could head to Heathrow the night before and stay at the airport or very nearby.

I've taken the coach between Oxford and Heathrow many times and there has never been any problem. Comfortable enough for the journey, very friendly and helpful drivers and frequent departures. There is no need to worry about the trip from Oxford to Heathrow.

They have group tickets that can be a good option for you, but I'm not sure if they are sold as singles or only as return tickets.

If the London stay isn’t set in stone, upon arrival go straight to Oxford. After Oxford, take the train to the Canterbury, and leave London for the end. That way, you could just hop on the tube to get back to the airport. If renting a car, go to Oxford first since it’s on the same side of London. That would make the day trips easier since you wouldn’t have to arrange transportation or tour. Again, if you had a rental car, from Oxford you could stop at Churchill’s house at Chartwell, and possibly a stop at Igtham Mote, en route to Canterbury. Another possibly day trip from Canterbury is Leeds castle. After leaving Canterbury return the car to the airport and take the tube to your hotel. Choices, choices.

I'm another on the side of Airline coach service.

Fast, frequent, inexpensive and very reliable. Much less hassle than being in London.

I've never used it but I used to regularly drive the route in my car. The M40 from Oxford to the M25 around London is usually very straightforward, and if you are awake enough to look out the picture windows the scenery is pretty good.

The M25 can be crowded and slow, but the Oxford Bus Company know their business well and they know how long the trip will take, and build in sufficient time. I'd recommend them with no worries.

Thank you so much for all the helpful replies! It sounds like the transfer from Oxford to Heathrow is easier than I thought.

I've nothing against the Oxford Bus Airline service but with 5 or 6 people you might find a (large) private hire car comparable in price (& probably a bit less) at £80-100 depending on exact numbers and be a door to door service. Personally I tend to use 001 Taxis but there is a large number of firms regularly doing this trip.

Thanks Marco! I will look into it.