Hallo Pip,
Welcome to Rootschat!
I cannot help you with searching for the graves, but I found the following death on
www.freebmd.org in the hope that the cemetery you are looking for is in East Glamorgan

William H Green age 74, Reg No: Dec Quarter 1936 E. Glamorgan 11a 943
Mary A Green age 48, Reg No: Sept Quarter 1941 E. Glamorgan 11a 1154
Of course, I am not sure how old your relatives were at the time of death. The one for a Mary A Green seems a little young to me because you have given a time frame of 1960-63, but that was the only one I could find that might fit.
There are a couple of deaths for a Mary A Green in Swansea, but they are either much earlier, or too young.
We can then take it from there.
Viv