Getting a minor wound bandaged up is not what A&E is meant for, it's for life-threatening injuries.
Going to A&E and waiting there also means you're losing 4 to 6 hours.
111 you just get some robot asking you a never-ending list of inane questions before someone tells you to either self-care at home or go to A&E.
A pharmacist should be able to administer the supplies they sell, particularly wound dressing and care. It's a requirement in some other European countries like France (where pharmacists are doctors), but in the UK the reality is that most are unable to do so.
111 do a lot more than that, they will get you a GP visit even if the GP claims to not have slots, and they will get medical professionals to come to you.
If it was in hours, I'm surprised they didn't get a nurse appointment for the cut.
What is wrong with going to A&E for an (as you said yourself) emergency?
A pharmacist dispenses medications and should know about their safe usage. They won't tell you how to bandage a wound.