Hi Nick, don't be put off by no headstone. Many people did not have them until quite late. My own grandfather is buried at St Giles (he died in 1957) and he has no headstone.
I do know that in many cemeteries and church yards headstones are not allowed on 'common' graves. This is a grave, purchased cheaply, and usually contains the remains of three unrelated people. I don't know when this fell out of fashion, but you cannot to the best of my knowledge have a common grave today. I don't know if my grandfather's grave is a common one, but I suspect it is. If there is no-one on this forum able to assist, there is another forum that may be able to help and that is the Birmingham History Forum. I no longer live in Birmingham, so I am unable to help, but the BHF is worth a try.
Billysgran
