Must add a bit of info here
If you are looking for details of from a specific cemetery, before contacting any local council, do yourself a favour and see if you can find the number of the Cemetery Site Office, in many instances these people will help with some information, such as whether there is a marker on the grave. But you will need exact year etc., and they can often tell you if this was a family plot.
I have had a lot of success in this direct approach, just be sure you have the exact information you need and be pleasant & polite (I would not expect anything less from anyone on here!).
So far the best ones I have contacted are Redstone (incorporating Reigate Parish Records), Kensal Rise (including the Catholic part), Mill Hill (that was during a visit). And a couple of churches, I never use a church website for this, but will phone or write (with an sae)., for some reason I have found that many churches just do not reply to any emails.