Thingwall is a bit of a mystery. Miss E Wade owned two fields on different sides of the quarry as did J Shaw. Miss E Wade owned a lot of land around and about. Part of my garden was from her estate but unadopted. There is a field off Galloper's Lane which she also owned, but after a recent effort to find who owns it now, all the locals came up with blanks. There is no apparent will, and therefore nothing can be proven. Vyner's owned Mill Farm and other plots in that location, from which there would appear to be a short path leading to the quarry.
Do you have a copy of Greg Dawson's book Tingvelle ? There is a map with field names at the back for 1850. In the book Greg Dawson says he used to play in the quarries as a child. He lives in Irby. You might find a contact number on the internet. His address is there. He has a lot of information.
There is also this document which might give some more info on land owners, not sure how much you have already. http://content.yudu.com/Library/A2qad1/HeswallMagazineFeb20/resources/16.htm