Your confusion here is totally understandable. It might help to explain the way burial plots are sold. When you buy a burial plot, often what you're actually doing is buying a Grant of Exclusive Right of Burial, which is the right to decide who is buried there for a set period of time (usually about 25–100 years). It's very much like purchasing a lease.
When the lease runs out, you'll be sent a letter to ask whether you'd like to renew. If you decide not to, the person buried in the plot won't be disturbed, but the headstone may be removed, and additional burials may take place in the plot (a little above, for example).
So, how is a burial plot passed down? If you became a joint owner of a plot during your lifetime (to do this, you’d need to have applied to the authority in charge of the cemetery) then the surviving owner (say, a spouse) retains the Grant when you die.
If you were the sole owner when you died, the Grant becomes the responsibility of the Executor or Administrator of your estate (if one has been appointed) or your next of kin (if not). If you had a will, they will transfer ownership according to your wishes. When there is no will, ownership should be transferred according to the laws of intestacy, or with the permission of all people who would have inherited if those laws were followed.
However, to make any transfer valid, the Executor/Administrator/next of kin still has to apply to the cemetery and get the paperwork in order, with a copy of the death certificate.
There’s a pretty good explanation of this here, for further information: https://www2.merton.gov.uk/community-living/bdmcp/cemeteries/graves-memorials/grave-ownership.htm#when