If you still have a European Health Insurance Card, it will be valid until the expiry date shown on the card. It is not valid anymore in Switzerland, Norway, Iceland or Liechenstein regardless of the expiry date.
Certain people will still be able to apply for an EHIC details of which can be found here.
The whole subject of health treatment is still being discussed and the UK government have signalled the introduction of the GHIC, the Global Health Insurance Card. The details of how the GHIC will work are not yet available.
Both the EHIC and GHIC are free to get from the government website by completing a simple form. There are also many websites that have been set up with the objective of providing these cards. These third party sites almost exclusively exist in order to charge you money or farm your personal details. These cards are free to obtain from the government website and you should avoid paying third parties for the service at all costs.
You are strongly advised to get appropriate Travel Insurance when travelling to the EU. It is not a legal requirement but the cost of getting treatment can be very expensive while abroad and the cost of bringing you home in the event of a serious accident can run into tens of thousands of pounds.
When applying for Travel Insurance you must disclose all of your medical history. Not doing so, even accidentally, can invalidate your whole policy. Witholding information deliberately, presumerably just to get cover or to save money, is fraud and as well as invalidating the policy you run the risk of getting prosecuted.