As Diver0001 said, in the UK you generally have two choices, unless you are in Scotland where there is also ScotSAC.
PADI is a commercial organisation. Pay as you go. You sign up for a course (Open Water) , the course will happen on the date agreed, in the time schedule agreed. The expectation is that you will move from course to course.
One big advantage is that you could do a referral, where the theory and pool (sheltered water) sessions occur in the UK, and then complete the course overseas, in a warmer climate, doing the open water session.
BSAC is a branch based system. The Branch you join provides the training. The instructors are all nationally trained and assessed. (It is possible to go to a BSAC School, which is a commercial facility - similar in structure to a PADI facility).
The big thing is that branches vary significantly, some are very focused on diver training, some are not interested. Some have boats, some don't. Some go diving, some don't!
You 'could' do a referral through the BSAC, but this is a little more difficult, because branches don't intend you to do this, so are not structured to recommend or support it. A lot of the cost advantage of being a branch member would be negated.
If you intend to learn with a branch interview them, make sure they have a number of ACTIVE instructors. Make sure they are the type of people you want to spend time with, make sure they go diving, and include diving for the lower qualifications.
Where a PADI course will be setout on a specific timetable, a branch teaches using volunteers. This can take a long time.
The PADI course will probably be two weekends, one weekend covering theory and the pool work. The second covering the open water diving.
BSAC branches will teach weekly, either both pool and theory each week, or alternating pool and theory. The open water dives will fit in with peoples spare time, and generally this is where things can slow down.
One big advantage with BSAC is that the people who train you are the people you will be diving with. A good branch means you always have a pool of divers to go diving with. The general philosophy is that you are always being taught and are learning.
You should be able to complete two qualifications in the first year - if you are proactive - if they are an active branch.
The starting qualifications are very similar - PADI open water, and BSAC ocean diver. The core basics of diving are the same, any way you slice it these fundimentals have to be taught. The variation occurs as you progress. BSAC qualifications intergrate rescue into the core training from the start, PADI teach rescue much later in its structure of qualifications.
Decompression diving is part of BSAC dive qualifications, you may or may not choose to do compulsory decompression and only do no stop dives, but you are qualified to do it if you wish.
BSAC is very structured to UK diving, its intention is to teach divers that can work as a buddy pair, independent of instructors or other members of the group. Seamanship , boat skills, rescue, etc are all part of the structure of BSAC training and course.
I am a BSAC instructor, I am also a lapsed PADI DM. I started with PADI, then joined BSAC. I hold qualifications with most agencies. TDI, IANTD, to name two.
You can move between the agencies. Most BSAC branches are more than happy to have PADI divers join. A PADI diver can either dive on their PADI qualification, or cross over into the BSAC qualification structure, as they wish.
There is a branch finder function on the BSAC website which might be helpful.
www.bsac.com
Gareth
PS our branch forum is here
www.sisac.co.uk/forum or the website
www.sisac.co.uk
If you have any specific questions fire away.