Could you easily get to Heathrow? Obviously there are all the major car rental companies there, and it’s a very straightforward drive to Oxford using the M40 then a short section of the A40.
Oxford has a large number of Park & Ride sites, including one - Thornhill - situated on the A40 itself.
You can leave the car there for up to 3 days, then continue your journey along the M40 to the Birmingham area.