So, one afterthought since visiting the car museum has been: do they have the kind of car Tolkien drove? The simple answer is that I don't know, because I didn't think to look up what I'd need to know to find out ahead of time.
Luckily, we know what kind of car Tolkien drove (and how he drove it) Carpenter's account
. . . in 1932 . . . Tolkien bought his first car, a Morris Cowley that was nicknamed 'Jo' after the first two letters of its registration. After learning to drive he took the entire family* by car to visit his brother Hilary at his Evesham fruit farm. At various times during the journey 'Jo' sustained two punctures and knocked down part of a dry-stone wall near Chipping Norton, with the result that Edith refused to travel in the car again until some months later -- not entirely without justification, for Tolkien's driving was daring rather than skillful. When accelerating headlong across a busy main street in Oxford in order to get into a side-street, he would ignore all other vehicles and cry 'Charge 'em and they scatter!' -- and scatter they did. 'Jo' was later replaced by a second Morris which did duty until the beginning of the Second World War, when petrol rationing made it impractical to keep it. By this time Tolkien perceived the damage that the internal combustion engine and new roads were doing to the landscape, and after the war he did not another car or drive again." (TOLKIEN: A BIOGRAPHY, p. 162)
So, he only owned a car from 1932 through about 1939 or so. THE TOLKIEN FAMILY ALBUM tells much the same story about the trip to Hilary's (suggesting that the Tolkien children were Carpenter's source for that bit), although with only one flat tire, but does not repeat the 'charge 'em and they scatter' story. They not only add a few details -- for example, that it was after their vacation to Lamorna Cove with the Wrenns that Tolkien bought the car (FAMILY ALBUM page 62). Since the visit was during the summer of 1932 (probably in August), and on at least one other occasion the family visits to Hilary was scheduled for around or towards the end of harvest time, that puts circa September/October a likely time for the new car purchase.
What's more, THE FAMILY ALBUM actually has a photo of Jo (or Jo II). It's labelled as "Jo II" but the license plate is that for the original Jo -- perhaps in England you get to keep the same plates when you get a new car? One other slight suggestion that this might be the original Jo is that the closet match I cd find online for the one pictured here is the 1933 model -- but then not knowing the date of Jo II makes it harder to find which model this might have been, at least in a casual online search. Here's a link to the closest match I cd find to the car shown in the FAMILY ALBUM.