IME midwives and health visitors usually don't know where the support groups are. I find this quite strange because it can make such a big difference, and there is usually at least one in most areas unless you are very rural.
I suppose that the NHS cannot promote them, because so many of them are run by groups with agendas other than simply breastfeeding support (La Leche League promotes an attachment parenting style, NCT wishes to recruit parents to paid groups and classes, IBCLCs and the Baby Cafés work for profit) - but there are so very few NHS run clinics, so it would actually make sense to distribute details of these groups even with a disclaimer.
Here's how you find the support groups, BTW:
Use the following search forms for the five most well-known BF support organisations in the UK:
www.laleche.org.uk/find-lll-support-group/
www.nct.org.uk/branches
abm.me.uk/find-a-local-breastfeeding-support-group/
www.breastfeedingnetwork.org.uk/sample-page/
www.thebabycafe.org/find-a-cafe/
Phone your nearest Children's Centre (their websites are often dire) or drop in, and ask if they have any BF support groups in their network. Bear in mind they might have strange office hours. Weekday mornings are usually a safe bet, but keep trying if they are closed.
Phone a Children's Centre in any other nearby areas and ask them too. Frustratingly the Children's Centres in neighbouring towns/cities often are not connected.
Find an IBCLC locally and contact them to see if they offer any groups, courses or classes, or if they know of any organisations who do: www.lcgb.org/find-an-ibclc/
And, yes, it's worth asking your midwife in case there is an NHS clinic in your area, but it's not that likely that you will find one.
As a last resort, you can also try googling "Breastfeeding support + town" with your nearest local towns and cities. This might bring up some lesser known options.
TBH, it should not be this hard.