It all depends on how ill you are (or think you are...)
Here are the options.
In short, if it's urgent, go to a walk in centre or A&E who will see you pretty-much immediately (as a triage), do whatever scans and tests are required, and if necessary admit you to hospital.
Otherwise for seemingly less urgent stuff, you can go and see your GP (you might need to make an appointment),
The GP may deal with the issue directly, or they might refer you to a consultant (in which case there may be a delay as you move up their list.) Or they might tell you to go straight to A&E.
But here's the thing - it'll be exactly the same consultant, and quite probably in the same hospital, as you would have seen if you had gone to A&E in the first place (if you were ill enough.)
In short, delays, when they occur, are mainly due to the triage process.
And there's always the option of going private (either paying directly or through insurance); you still have to go via your GP (for non-A&E issues) but you can jump the consultant's list. (Yes, the same consultant, but this time in a private hospital for a day or two a week.)
There's no benefit to be gained from private health insurance for genuine emergencies; if you had a heart attack in a private hospital they would take you to the NHS A&E. But you might have a nicer room, and better food.