It can not be that difficult to trace the owners can it? Have they not heard of Land Registry & surely the local council granted permission for the quarry in the first place.
...Lister;~)
It's nowhere near that simple. The quarry started in the late 19th century and was run by one of the firms who amalgamated in the early 20th century to form the Buxton Lime Firms. They in turn were bought out by ICI in the mid-20th century, and later ICI sold out their limeworks division to Tarmac. This did include some of the disused sites but some bits had already been sold off AFAIK, such as the ground the HSE bang lab is built on and the ground the WW2 bomb store is under. Other parts of the site have been redeveloped as a business park in modern times, and there's a speedway track round the back
As a result, the Harpur Hill area is a patchwork of ownerships and there are several bits of apparent no-mans land there, and the blue lagoon is one of these. You could try tracing ownership forward from the BLF but you'd struggle to show whether ownership was still with whatever company inherited the remains of ICI's lime division, had been passed on to Tarmac, or was with someone else.
To be quite honest, the only real option the council have if they want to take action is go for a compulsory purchase order and put notices in the press inviting anyone who thinks they have a claim to ownership to come forward, which is pretty unlikely under the circumstances!
Follow the horses, Johnny my laddie, follow the horses canny lad-oh!