To be fair Rotherham was the nearest town where I grew up (Bolton on Dearne), 8 miles to Donny and Barnsley, 7 miles to Rotherham. we always went to Donny or Sheffield shopping etc. We never went to Rotherham - except to Millmoor when The Reds played there.
You can try 111 - I did that when I thought i had sprained my wrist and they told me I had to talk to my GP within 24 hours. Armed with this info I phoned my surgery and told them what 111 had said so no choice but to get one of the doctors to give me a call - Phoned back that day and arranged an X-ray at our local health centre - Radiologist took one look and sent me to A&E with a confirmed broken wrist It was a lot better than just turning up at A&E as they already had X-rays so I jumped a lot of the Queue
It's weird once you're already in the system it works quite well. We took George last year and once he'd gone through the wait and was finally admitted onto the Children's ward they couldn't do enough. Even when he was discharged the Doctor on the ward old us that any changes to bring him back to the ward and saw to it we got additional contact details.
Call or visit your doctors and say you're worried you might have skin cancer. They'll sit up and take notice then, fit you in quickly, probably take pictures and send them off requesting an urgent 14 day hospital appointment for you. If you can't wait that long, get an appointment with a local Spire or BUPA clinic. I had such a situation 6 years age and rang my local Spire, got an appointment in 3 days and had it removed the following week. OK it cost me a grand - £100 consultation and 900 plastic surgery but they removed it (it was a cancer) in time and I'm still here.
Needed an appointment for my son yesterday, phoned that morning, got an appointment for 10.30am same day, got there, seen at 10.35am, out by 10.45am. It can work, the fact it doesn't for all is probably down to bad practices and lack of funding.