My homecountry CAA has a list of approved ELP examiners, from who they'd accept verifications.
What if you arranged beforehand to undertake the exam at a checking organization abroad, from who Hungarian CAA will accept the results? The logic being: if orgaization X is accredited examiner for i.e. German CAA, would HCAA be willing to accept their assesment of your language competence to issue the ELP onto the licence?
Anyways, as you specifically ask about the UK, here's something from their website:
https://www.caa.co.uk/General-aviati...e-proficiency/
the link to STD 31 does not work, try this one:
Standards Document 31: Organisations Conducting Approved Courses of Flight and Ground Training
(keep in mind that UK is specific, as they actually speak the language).