There are copies of the Parish Registers which include the burial registers for Llantwit Juxta Neath and for Neath (which I think are the for new cemetery ) on the shelf in large books to look at at the Swansea Archives. I looked at them this week before visiting the cemetery (the cemetery on the opposite side of the road to the council cemetery).
Unfortunately though there is no burial plan, so if the grave is not marked with a headstone you probably will NEVER find it. There are also a lot of graves just above the church in a jungle of Japanese knotweed that are innaccessible without a machete!
It's worth checking with the GFHS to see if they have transcribed the graves.