This is more complicated than it may at first appear, and there is unlikely
to be a simple answer. However...
Burial plots are not, for the most part, owned. They are rented (on a very
long rental term) from the burial authority. This may be either the church,
or the local authority (either a town/parish or district/city/borough
council). After a certain amount of time, if the plot is no longer being
actively maintained, the rights revert to the burial authority. But if the
deceased has living relatives, then whoever paid for the plot will still
have the right to maintain it.
The headstone, however, is the property of whoever paid for it. Again, once
it has been there long enough, if the grave is no longer actively being
maintained, then the headstone is assumed to have been abandoned into the
possession of the burial authority.
In your case, though, neither of these will have occurred as the grave must
be a fairly recent one. So the short answer is that the burial plot is still
in the possession of whoever paid for it, and the headstone, equally, is
still the property of whoever paid for it.
If that was you, then they (whoever "they" may be) have no right to remove
the headstone. But if the headstone was paid for by someone else, then that
person has the right to remove it.
Where it gets more complicated is if the burial plot and the headstone were
paid for by different people. In that case, although the headstone remains
the property of the person who paid for it, it cannot normally be removed
from the grave without the permission of the person who has possession of
the plot (which will normally be the person who paid for it).
I should add that, unfortunately, disputes like this are not uncommon. As a
first step, you should contact the burial authority and ask for their
advice. If it's a church graveyard, then the vicar is the appropriate person
to contact. If it's a town/parish council graveyard, then contact the
town/parish clerk. If it's a district/city/borough council, then you may
need to track down the relevant department via the council's website. But
the one thing you can be sure of is that this won't be the first time
they've had to deal with a case like this!
If you are the person in possession of the plot, and you paid for the
headstone, then the burial authority will normally write to the other party
to remind them that they have no rights over the grave. In most cases,
that's enough to end the matter. If not, and you are determined to pursue
it, then you will probably need professional legal advice.
On the other hand, if you are not the person in possession of the plot or
the person who paid for the headstone, then they probably do have the right
to remove it. This is why, if you are at all unsure, you need to get the
advice of the burial authority.
Mark