OK, history lesson: LLL was the first lay organisation of breastfeeding counsellors. Their British wing split about 20 years ago, and so there came to be LLL(GB) and the Association of Breastfeeding Mothers. ABM happened because some LLL(GB) people did not appreciate the US trying to tell 'em what to do.
NCT began in the 50s and about 40 years ago started to train breastfeeding counsellors as well as antenatal teachers.
About 15 years ago (I am not sure of the dates) the wider organisation of NCT (not the breastfeeding counsellors) wanted to take sponsor money from Sainsbury's who at that time had their own branded formula. It didn't happen, but as a result of the discussion about it, many breastfeeding counsellors left to form the Breastfeeding Network. BfN will not take money from anyone with an interest in infant feeding (inc retailers, nipple cream people, bras, pumps etc) . NCT are not quite so strict (though many breastfeeding counsellors would want us to be) mainly because other parts of the membership don't feel as strongly.
There are no particular rivalries between any of the organisations - we often work together locally and nationally. Obviously, by now, the bulk of all the memberships and workers were not even around when the 'splits' happened. It's not a big deal.
Of course there should be one number. There was going to be a few years ago and it fell through (not because of politics, more practicalities). There may be, eventually.