I'm a bit late, but welcome Kieron.
Like you I lived in a coalfield area and was inspired to explore non-ferrous mines - thanks to Dick Bird's pictorial books (newly published way back then) which showed not just historic photos but also photos of people exploring disused mines 😮 So I started off exploring and surveying abandoned coal levels; by a mix of luck, caution and reading around, I survived. In due course I found myself with wheels and living within reach of non-ferrous mines and slate quarries which I explored with friends from a caving club that had a strong interest in mines as well as caves.
If you lack wheels and have difficulty in rustling-up friends to go underground with, you could do worse than start off with a caving club (you said you were interested in both mines and caves). Caving clubs and independent cavers are more numerous than mine explorers so this kight be an easier way to start off. The main thing is to get used to the underground environment (whether natural or mined), try different equipment, and get a feel for things. Most cavers (especially those that cave in Derbyshire) do at least some mining. There must be some club members within reach of wherever you live in South Yorkshire who could give you a lift to meets, whether in North Yorkshire (or maybe mines in Northumerland or Cumbria) or in Derbyshire (where there is a good mix of mines and caves - often inter-connected).
As other posters have said up-thread, Derbyshire is probably the best and most convenient area for you to start in. I'm a member of a Derbyshire club and have done a moderate amount there. The inter-connecting caves and mines (some Derbyshire caves were discovered by mines breaking into them) are a specially interesting feature of going underground in the region, and is not often found in other regions of the country. Indeed, it would be difficult in Derbyshire to limit yourself to only mines, or to only caves, as so many are inter-connected.
Most caving clubs have mine explorers amongst their members - just ask how much mine exploration the club does before joining. Most caving clubs run a couple of mine exploration tripss each year - often to Cumbria, Northumerland / North Yorkshire, Wales or Cornwall - all of which are classic areas.
You may get an offer of some trips from this website. You could also introduce yourself on the uk caving website and ask whether there are any clubs in your area that do a reasonable amount of mining [Edit: I see you've already enquired on uk caving]. There is a good list of clubs, with contact details, published in 'Descent' magazine which appears every two months. The NAMHO website has a list of mine exploration clubs (and caving clubs with an interest in mine exploration) with links to those clubs' websites. Many club websites contain lists of meets so you can gauge how much mine exploration they do, and what regions of the country they tend to visit.
So my advice is to get underground with whatever cavers or mine explorers live closest to you, examine and try out various equipment, and get a general feel for the underground. And if its in Derbyshire you can't fail to visit non-ferrous mines.