Competition works when there is genuine choice. Health is more akin to a train than a shop - You don't have a choice of the service in your areas, although there may be multiple providers across the UK - if you need emergency treatment, you go to the nearest A&E.
Maybe if the system is fixed similar to NICE with drugs, so that there is a set rate the NHS will bill, and its then private insurance who don't have a stake in the hospital or a service locked like in the USA then its possible an end user could get a good premium for insurance, but that wont improve the hospital itself.
What you would likely see is certain hospitals which have funding would become more popular, creating waiting list there, and other hospitals in the area would not be as profitable, they would then have to cut services, driving more people to the other hospital creating vicious spiral. Other hospital could then up prices knowing they have greater market share, meaning we end up in a system which has unequal outcomes.
I am not against private business being involved in the NHS but they should be for driving down costs of support services and not in healthcare provision (unless we go with private hospitals but funded by the state for all patients so the end user doesn't see the difference) but if we do this then privatisation and capitalism has shown where the costs are fixed they will then drive down standards to increase profits, so we will ultimately end up paying more of our tax fees, to a lower quality service, only for it to end up in a tycoons remuneration pot