Hi,
As mentioned , you do need permission from the relevant council.
I would tend to think each council probably have different rules regarding their cemeteries; so I would phone the cemetery department in the relevant council and inquire about the grave that you want to put the headstone on . They should be able to tell you if you can or cannot put a stone on it, and what steps need to be taken if you can.
My mum-in-law has very recently ( about 4 months ago) put a tablet/plaque on her unmarked grandmothers grave in Swansea who died in 1912 (which is not a privately owned grave, but there are no non family members who are buried in there with her - we have a list of who are buried in there from the council)
But I must add that my mum in law was told when she asked in the church on the grounds that she needed to apply for permission from the council, but she never did apply for permission from them .
She just ordered the tablet/plaque and what she wanted engraved on it . She then drove to the graveyard with her daughter and the plaque in the boot. Then she got the poor daughter to lug this stone plaque (which looks around about 2ft by 2 ft looking at the picture my mum in law has sent to me) to the grave and plonked it down on top of the grave without asking anyone's permission - That's my mum-in-law for you!!

They will probably remove it if anyone does notice that it should not be there( which is probably doubtful that anyone will notice it though as it's a big graveyard )
Why she didn't seek permission is beyond me, as If they do spot it and remove it I bet she'll have a good old whinge about it.