There can be many reasons for exhumation, for example the family has moved away and wish the remains to be in a cemetery near where they live.
A licence from the Department for Constitutional Affairs and/or a Church of England Faculty will be required, dependant on whether the remains are buried in consecrated ground. This applies to both the exhumation of a whole body or cremated remains. The application should be made by the deedholder to the grave.
There are specialist companies that carry out this work and any undertaker will advise you.