If the property is registered in the Land Registry (you will have a folio number), then the original deeds are stored permanently in the Land Registry and all you need as proof of title is a certified copy of the folio.
If it is not registered in the Land Registry, it is "unregistered land" and you will need the original deeds to convey it on to someone else. To address your specific concern, I believe the Law Society assists in the closure of practices and ensures that important documents are not destroyed and can be found (as much as possible). If the deeds do get lost, it's not the end of the world as title can be reconstituted through copies, Registry of Deeds records etc, although this will cost a few quid and probably take some time. Defective title insurance can also be obtained.
If the land is unregistered and you have a mortgage, the bank will have them. Othewise, most people leave their deeds with the solicitor, but there are some who want to hold on to them.
If you are thinking of selling, make sure that you have access to the deeds before you put it on the market, especially in the present circumstances as buyers will be on the lookout for excuses to withdraw, or reduce their offer.