Hello Robert,
I grew up very near to the Cowley Road Hospital. Were you are aware that it closed around 1983, and are you able to visit the Oxfordshire History Centre where the surviving records are now stored?
The nearest Anglican church with a burial ground is St Mary & St John. I have a CD from Oxfordshire Family History Society who transcribed some of the parish registers, but unfortunately the burials stop at 1929. Ancestry have now released most of the county's PRs online so you might find something there. If you're not a subscriber to that site, I could have a look for you on Tuesday 29 November when I'll be visiting our local library.
There were also municipal cemeteries in Headington (closed 2003)and at Rose Hill (closed earlier, I think). Again, the CD transcripts stop well before 1965 and these records aren't online, so you could try the History Centre for them as well.
Oxford Crematorium opened in 1938 and is still in use (I was there in June for a family funeral).
Presumably you have the death certificate - where was she living before she went into hospital?
I hope this will give you some ideas. I have to go and eat now but will try to think of more suggestions.
Carol