Some graves were purchased in perpetuity and those would be in a different catergory to the normal leased graves, however the great majority of graves are in the latter where the BR's last a specified term. At present only the London boroughs are generally re using graves [example City of London Cemetery] as new regulations would be needed to be introduced by parliament for the remainder of England and Wales.
In cemeteries under the control of local authorities, there are existing powers to add additional burials where a grave say was purchased as a double and only one burial took place and it has remained so for many decades. Similarly in theory where BR's have expired the LA would be able to add additional burials. In theory this would equally apply to vaults and tombs. However many Victorian cemeteries have Listed buildings, graves of historic interest or boundary walls/gates. A ruling by a Consistory Court on West Norwood Cemetery ruled that the curtilege of a Listed grave applied to the whole cemetery. This ruling was specific to that cemetery but it has never been appealed or challenged. This means that any demolition etc would require LBC from the LA [or EH if grade II*]. Therefore LA's may now be more reluctant to interfere with existing vaults/ tombs and there is a "jobsworth fear" among many local councils that relatives of somebody who died 150 years or so before will sue for criminal damage or trespass if the council repairs or allows a voluntary conservation group to repair a vandalised grave. Using the same logic theses jobsworths may be reluctant to interfere with a tomb/vault in case the council were to be sued. Another factor is that many cemeteries such as Southampton Old Cemetery, Lister Lane, Leeds and Ford Park Plymouth are on an EH Register of Historic Parks and Cemeteries and although not quite so strong in law as a Listed Building, EH would surely place pressure on LA's if they began destroying historic or attractive graves so breaking the character of a cemetery.
The answer in brief is yes vaults/tombs etc would be treated as normal graves but many of these would be in consecrated ground or even under the floor of a chapel and the LA would need a Faculty from the Consistory Court.