they are in some circumstances already taking weekend appointments. This is easiest if it doesn't require a senior consultant in attendance. I've attended a hospital appointment at 7pm on a Sunday night .... it came as quite a surprise
They can use private resources when they have free slots, but again, i believe they are already doing this. I've always been seen in nhs facilities so i can't say for sure, it probably relates more to the treatment required.
I hear all these bad things about the NHS, and my experience is that GP surgeries are a mess, but, i have found NHS hospitals and staff to be quite wonderful and i have attended 4 hospitals in my region.
It very much depends on your ailment I think, but I can't speak highly enough from my personal experience