OK - a 'lactation consultant' should have done her International Board of Certified Lactation Consultants (IBCLC) exam which leads to this internationally-recognised qualification. At a guess, I would say there are about 50-100 LCs qualified in this way in the UK. Most (not all) are also midwives or health visitors, and some are also breastfeeding counsellors with one of the four vol. orgs in this country. A handful work in private practice as LCs, but the majority of the health professional LCs work as midwives and HVs doing all the things midwives and HVs do, though a few have infant feeding co-ordinator specialisms within the health service.
Anyone can call themselves a lactation consultant, as the term is not protected (though you cannot call yourself an 'IBCLC'). To sit the exam for IBCLC you have to have had very extensive experience academically and practically studying and supporting bf (see \link{http://www.iblce.org/become.htm\here}); several thousand (yes, thousand) hours, documented, of helping mothers if you don't already have a degree in some related field (when the hours are somewhat reduced). You do not have to have breastfed.
To become a breastfeeding counsellor is more of a mother-to-mother pathway. NCT bfcs typically take about 3 hours to train. Other orgs (LLL, ABM, BfN) have slightly different requirements, though I don't think the training is substantially shorter. You have to have breastfed - for NCT training it's six months.
Unless a midwife or health visitor has undergone IBCLC certification, or training with one of the vol orgs to be a bfc, or has had a lot of intensive and extensive post-qualification training, just about any bfc will know more about the social, technical and physiological side of bf (sometimes a lot more), and should be more skilled at the counselling side of it all, too.
There are a lot of short courses available which train women in a short time (typically, about 12 sessions) to be peer supporters/peer counsellors. This is a great way of harnessing keen breastfeeders to offer friendship and support to women in their own communities, but it is not sufficient to really get to resolve difficult long-term challenges, and without more experience, they really are not able to resolve positioning and attachment issues (apart from the more obvious ones).
It's totally understandable that people get confused by all these different qualifications. I got a call recently from a mother who told me she had seen an NCT breastfeeding counsellor who had been unable to help her in any way with her positioning....I was a bit concerned, but it turned out the 'counsellor' was a peer supporter.