We will be spending two weeks in England in October, flying in and out of Heathrow. Our first city stop will be Cambridge and our last overnight will be in Oxford. We will rent a car for most of our trip. Would you recommend renting the car and dropping it off at Heathrow, or taking public transport or shuttle to Cambridge, renting the car there and then dropping it off in Oxford to take the shuttle to the airport?
Assuming you’re coming in off a long haul flight what you should absolutely not do is collect a car from Heathrow on landing and drive any more than a short journey. Much better to get somewhere to get over jet lag and then hire a car. Assuming the one way rental costs are not too high dropping at Heathrow is perfectly sensible.
In addition to Johnew52's very accurate comments about jet lag from an
overnight flight, you'll be driving on the opposite side of the road (just an
assumption).
I also think dropping the car off in Oxford as you mention is a viable option.
It should be easier to drop the car off in Oxford, and you will save a bit of
time in making your way to the rental car area at LHR, then having to wait
for the rental car shuttle to drop you off at the terminals. Of course, if the
dropoff charge is less for LHR than Oxford, you might rethink, but you
should be able to get the costs upfront, and there is of course the cost of
the bus.