You might also think about why you want to get into cave diving. Is it the lure of exploration of new cave, visiting passage already explored but only open to divers or is it the excitement of diving in an overhead environment?
The reason I ask is that if it is the latter, you may find some cave diving abroad more appealing. There is a world of difference between clear water, floating amongst stalactites in the Yucatan and squeezing through passage in zero vis in the UK. In the UK, I believe cave diving is more a means to find new dry passage than an enjoyable past time in itself. Yucatan on the other hand you have stunning systems entirely submerged.
You will find that many of the techniques and equipment configurations vary depending on the nature of the passage. In the Yucatan you'll find the majority of dives being conducted using double back mounted tanks manifolded with an isolater valve. In the UK side mounts are the norm.
I have a limited amount of experience with cave diving in both environments (though I wouldn't profess to be an expert in the field). I started diving around 17 years ago in the open water - first with a padi try dive course and then back in England as a BSAC club member. I started caving about 11 years ago. I had a brief dabble with diving short sumps (50 m or so) using twin side mounted pony tanks and then took part in a cave diving course offered by the NSS CDS (US national speleaological society cave diving section) in the Yucatan. This course was aimed at teaching skills relevant to diving large passages in clear water using a buddy system with back mounted tanks and required (I think) that you were at least PADI advanced open water with 150 dives under your belt. Whilst I've heard plenty of people dismiss these courses as commercialisation of a risky pursuit, the reality is that they are professionally run and teach valuable skills that will improve your open water diving as well. The course included line laying, multiple line navigation, blind cave exits through constricted passage whilst air sharing (using a long hose regulator), relocation of the line in zero vis, etc. I would never assume this qualified me for cave diving in the UK as the skills are quite different but as a skill set that would open up guided tours of stunning cave systems with more experienced divers I found it well worth doing. If I planned to get into diving in this country I would contact the CDG - no question about it.
Regarding BSAC and PADI, years ago PADI had a reputation held by divers in this country as an organisation that valued commercial success over proper training. Jokes used to run about the acronym standing for "put another dollar in" or "pay and dive instantly", or for the even more cynical "pay and die instantly". PADI courses were run over a couple of weekends whereas the BSAC equivalent tended to take longer, being run through club training nights generally held once a week. The reality nowadays is that there isn't a great deal to call between the courses offered. BSAC have in many respects tried to emulate PADI to improve their own commercial success.
Of far more importance is the relevance of your experience. If you take a PADI course in florida, you won't find it hugely relevant to diving in Britain where the conditions are generally colder, stronger currents, bigger waves, lower vis. If on the other hand you train in Britain, PADI courses will prepare you for British conditions just as well as BSAC. When I trained with BSAC the training took place for an hour once a week in the local pool. It was several months before I had completed enough training to even get in the open water. Open water experience is what will make you a better diver and to this extent a course that gets you in the sea faster is a good thing.
So to sum up an overly long post
If you are interested in joining exploration in this country get in touch with the CDG and get to know some cave divers as all the other posters have suggested. If on the other hand you are more interested in the odd holiday to florida and the yucatan to dive in some stunning passage in the company of an experienced guide, you might consider my comments above. Regardless of what you do, open water experience will help your confidence and knowledge of the equipment and the physiological aspects of the sport - and is damn good fun in its own right!