Is there anywhere near the Cotswolds that one could coach to from Heathrow and where one could pick up a hire car for the last bit? Bath, maybe?
Last trip on the day I arrived I took a coach to Taunton, picked up a car there and then drove to Lynmouth. I stopped a lot along the way to take pictures and arrived quite late.
In April there is only a 1 hour time difference between South Africa and the UK, so there is no jet lag involved. I usually sleep pretty well on planes (I usually sleep pretty well almost anywhere ;) ) so I tend to arrive at Heathrow feeling fresh and raring to go.
A possible compromise might be to drive straight from Heathrow to Donnington, park the car and explore Stow-on-the-Wold on foot until we can get into the cottage.

"<i>A possible compromise might be to drive straight from Heathrow to Donnington, park the car and explore Stow-on-the-Wold on foot until we can get into the cottage.</i>"

That would be the best solution of all. Or you could also pop over to the Slaughters or Bourton or Burford. Plus you'll want to do some grocery shopping in Stow as well.

LHR is about the best/easiest place to collect a car. 2nd best would be in Oxford -- but that would involve a train(s) and/or bus to get to Oxford and then a taxi to whichever rental company you choose.

There really isn't anyplace else that makes sense.

There's a bus every half hour (no booking really needed: there's a slight saving if you do, but hardly worth the hassle of worrying about missing the one you've booked) from T5 and from the central LHR bus station to Oxford. You then probably need a taxi: the most driver-friendly car hire location is Hertz at Kidlington, about 3 miles from the city centre. Avis and Budget both have depots a half-mile, OK-ish, walk from the main bus station, but both then require you to deal with central Oxford traffic.

The other more or less Cotswolds place with car hire and painless access from LHR is Bath, though buses from LHR are every 90-120 minsish.

The railway station handiest for Stow is Moreton in Marsh, though car hire there is tricky, there may well be line closures this coming March or April (meaning the "train" to it turns into a bus at Oxford), and to get to it means a bus from LHR to Reading station or the Heathrow Connect train to Paddington

Thanks for the help, folks :)
janisj: good tip about the grocery shopping. Yes, we will definitely need time in Stow to stock up.
flanneruk: thanks for the bus info. I will check out car hire companies in Oxford and Bath. Last time when I took the bus to Taunton, Enterprise agreed to pick me up at the bus station and drive me to their offices. They even dropped me back at the bus station at the end of the hire. Perhaps I can persuade them to do it again :)