There is no automatic entitlement to private health insurance. Any private health insurance benefits should be clearly stated in your contract of employment as an express term.
However, irrespective of what the contract states, it may be possible to establish an entitlement by virtue of custom and practice.
Also, consider whether you are being treated less favourably than other workers for a reason connected with a characteristic protected by equality legislation, such as age or sex. For example, in the circumstance where private healthcare is only offered to longer-serving members of staff who are overwhelmingly aged over 50 or male, you might well be able to lodge a claim of unlawful discrimination with an employment tribunal.
The provision of private health insurance (except for overseas employment) will be considered by the HM Revenue and Customs as a taxable benefit.