The best advice I can give is to get involved with club activities. As you hail from Cumbria, CAT (Cumbria Amenities Trust) is probably your local best, or alternatively Mines of Lakeland Exploration Society is not too far away from you. All the clubs are affiliated to the National Association of Mine Research Societies who has an annual field meet. These field meets are more 'Indiana Jones' than a lot of old fuddy duddies and usually put on a variety of underground trips ranging from easy walk ins to advanced STR. The thing about the NAMHO field meets is that there are trips laid on into places subject to special access arrangements.
There are quite a lot of sites in Cornwall, many of which are sensitive and also many of which involve either SRT in some form or neck deep water, not to mention cliff paths requiring the skill of a mountain goat. NAMHO field meets down in Cornwall reoccur every few years and are generally well attended.
There was, and probably still is, a great little book "Mining Sites in Cornwall and South West Devon" by Barry Atkinson. If is out of print, try everyone’s favourite bookseller; moorebooks, he posts to this site.
Off the top of my head, you want easy walk ins?
Well there are several around Carbis Bay which you likely already know about. A wet feet, and possibly thighs behind the old boat house on the left of Cape Cornwall. (St Just United), hope you have proper lighting, there is a drop into a stope at the end of the entrance passage! While you are in that area take a look at the Bunny. If you follow the cliff path from Truthwall Lane towards Botallack Crowns you will find it. The upper workings look like a small quarry, and the adit is below.
Try the little book, half the fun is in the discovery!
My avatar is a poor likeness.