I guess it all depends on how generous you are with the definition of ‘young’ but I am assuming you mean people under about 30-35?
As Badlad has pointed out, NPC has seen a substantial influx of younger members in the last few years, many (but not all) originating from York University, Imperial College and Manchester. In fact, a significant number of the Committee would fall under the ‘young’ category including our chair, social secretary, meets secretary, tackle master, and a couple of our ordinary members too.
There is usually a fairly good balance of ages at our members' weekends these days (first weekend of each month), with a strong contingent of younger cavers, plus those in 40s, 50s, 60s and 70s.
We have just returned from our annual spring holiday in France, attended by about 30 people including at least eight members under 30 that I can think of.
In reality, I don’t think I have ever felt more positive for the future of NPC and other clubs.
I think other clubs around the Dales have seen a similar influx lately. I know the Craven did a top notch job a few years ago at getting themselves out there on social media and opening up their cottage to visiting clubs, and as a result have seen lots of younger members.
I guess if you are looking for a club where the balance is tipped more towards younger cavers (rather than a mix of all ages), I can't think of any that fulfil this other than the university clubs.