Hi, I now live in South Africa but spent my youth in Derbyshire and lived in Sheffield. Has already been said, there are many Lead mines within 30 miles of Sheffield in the Peak district. As an old retired Caver I am in a similar boat.
You could always go to Casleton and go down the Speedwell Mine. It is a "show cave/mine" and you will have to pay the required fee, but it will get you underground safe and clean and you will get an underground boat trip thrown in as well.
Regards, Trogladike. :thumbup: