the different BF organisations seem to do things in very different ways. The course your hospital/HV team are talking about is probably run by a counsellor from one of them (likely LLL or BfN) and would qualify you as a peer supporter. If you then wanted to take it further, you could do a full BFC course, which takes much longer & is more in depth.
I've done a year of training to become a BFC with the NCT. The NCT don't do any kind of semi-qualification - straight onto the BFC course or not. The course is basically a degree - they estimate at around 3 years to complete it, after which time you'll have a recognised university qualification. You find a local tutor with space and then attend monthly tutorials where you practise counselling skills and learn from each others presentations, and in the meantime the onus is on you to do a lot of reading, writing essays, preparing presentations etc. There are also residential workshops at a couple of stages. I really, really enjoyed the tutorials I went to but found it hard to create the extra time needed to push on through the course. I also found it very, very theoretical.
I couldn't commit to getting to my tutorials anymore (there weren't any local tutors, so mine is over an hours drive away, which is too much for me with 2 children) so I haven't re-started after the time out to have Eli. There is an opening in a tutorial group closer to home so I might start that in Sept, or I might start again completely with the BfN.
With the BfN, I'd do a peer supporter course first, which would be 8 consecutive weeks worth of a 3 hour course. That would be run out of a hospital, and I'd then be expected to start up voluntary work as a peer supporter within that hospital or hospital trust (I've been told they'd like me to commit to twice a month at least). If I decide to continue to train as a BFC with the BfN then tutorials are in the evening (suits me much better with childcare). At the moment I'm leaning towards starting from scratch with the BfN - the course itself seems a bit more hands on and I like the idea of being kind of pushed to do voluntary work in conjunction with it rather than essentially getting a degree in counselling.