I don't think the NHS has had an overhaul since it began but how many people will want to" dip deeper " the only other thing is privatisation but nobody wants that either ,so what's the answer.
Higher taxes?? I don't think it has to be private. Canada's healthcare system is universal and is good but we pay way higher taxes towards it.
Edit: We pay more for prescriptions (free for seniors), optometrists, dentists, ambulance (around $400 w/out coverage), etc. Many companies offer good health insurance through work for that kind of thing. We get almost full coverage (rare) for the dentist which is amazing, as otherwise cleaning would be well over $1,000 (not private, going by the provincial fee guidelines). So paying some amount for those things will help the system too. It is not like the USA though, where if you do not have coverage and need an ambulance, they will not pick you up. They will, you will just be billed for it.
Everything else is covered through our provincial healthcard (hospital, doctor appts, bloodwork, imaging, treatment, etc).
So I would say the Canadian healthcare system is somewhere in between the NHS and USA systems, and overall works in regards to how great the doctors, treatment, hospitals, etc are. Things are a bit different in Quebec due to corruption.