I'm not sure if they would offer treatment for an existing health problem.
What Benenden offers is a service for certain conditions if you cannot be seen quickly by the NHS. Even then you could have a scan at a particular hospital but you cannot have the necessary procedure carried out there so have to start over again at another hospital.
It can be very useful but it works in addition to the NHS not as a complete private healthcare system.
You would usually have to contact your GP first, then, if NHS waiting lists proved too long, approach Benenden.