Interesting stuff. I've been going to that area since I was a nipper and don't recall seeing that lift. In the 60s, Hades Night Club was underground at the Lido. We supplied some of the equipment. It had a great atmosphere with all the disco lighting but when the service lights were on, it was a dirty tip of a place.
This photo shows the Lido in its prime. The lift can be seen on the left, partly hidden behind the cliff.