As I have argued before it is up to the owners of a building what happens to the building. The owners have the property rights over the building, which include the right to bring it down, if they so wish, and the Great Christchurch Buildings Trust - the group wanting to stop the cathedral from being demolished - gets no say in the matter. The news item points out that the Court of Appeal has dismissed all the arguments made by the Trust.
Oliver Wendell Holmes Jr. made it clear as to what ownership means,
But what are the rights of ownership? They are substantially the same as those incident to possession. Within the limits prescribed by policy, the owner is allowed to exercise his natural powers over the subject-matter uninterfered with, and is more or less protected in excluding other people from such interference. The owner is allowed to exclude all, and is accountable to no one. (The Common Law, p193, (1963 edn.))The Anglican Church has these rights and their rightful exercising of them means that the cathedral can be demolished, if they so determine it should be, and that decision is to be "uninterfered with". The Great Christchurch Buildings Trust, or any other such group, must be excluded "from such interference", the Trust has no right to try and usurp control over the cathedral away from its rightful owners.