We're lucky with our GP practice. If you say 'non-urgent', you'll get an appointment in about 2 weeks. If you explain that it is urgent, they will find you an emergency slot after a few questions.
E.g. I started with a pink mark on my shoulder one Friday, which was really painful. It worsened over the weekend. I rang 1st thing Mon. as I was beginning to feel wobbly and the pain was intensifying.
I got a same-day appointment with the nurse-practitioner, who agreed with my self-diagnosisis of Shingles and issued an immediate prescription for anti-virals (available from the pharmacy on site). I was still pretty ill and have only just recovered, but the anti-virals helped immensely.
All GP surgeries should be able to operate like that - or even better.