172510, you need to read the CAA FAQs more thoroughly.
It states:
Part MED allows for a licence holder to obtain a medical certificate from any Authorised Medical Examiner or Aero Medical Centre in any State, but when they do so the record of the outcome will be sent to the State that holds their medical records, which will be their State of Licence issue. (FCL.015 and MED.A.025 refer). As a medical certificate must be obtained before flying solo, the State of Licence issue for an individual will be determined, at least initially, by where that first Medical Certificate is obtained.
Consistent with the above, from 1st July 2012 the CAA will not accept applications for licences or ratings from anyone who does not have their medical records held in the UK.
The answer therefore is yes, you can get your medical renewed in any Member State, but you must have it sent to your State of Licence issue. It may also need to be translated into the national langauge of the State of Licence issue, which could involve extra cost.