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.