I am a UK resident and while visiting Switzerland I went for a health check. I have a EHIC and wonder how I can apply for a refund. On the EHIC website it says that one can apply for a refund but I can't find an actual form or any information where or how I can apply for a refund.
If you have paid upfront for an eligible service under Swiss compulsory insurance law (KVG/LAMal), you will need to send a claim to Gemeinsame Einrichtung KVG, with your bill, EHIC, bank information along with possibly other required documents.
You will need to pay a patient contribution depending on the length of your treatment. Currently the amount is
The lump sum amounts to CHF 92.00 per 30 days for adults and CHF 33.00 for children up to the completed age of 18.
Generally, a physical examination motivated by specific concerns are certainly an eligible service, while a general routine checkup is not necessarily one depending on your age, medical history, the services provided etc. You can ask the doctor/clinic if the service is an eligible one.
I was unable to find a link but IIRC there is (literally) an app for that, with country-by-country information. Since you are from the UK, you can also refer to this page from the NHS (especially “If you're abroad and do not have your EHIC with you” and “Claiming a refund”). There is a phone number you can call.
Note that while it is sometimes possible to fix things after the fact and get a refund, this is not how the EHIC is supposed to be used. It doesn't necessarily cover “health checks” either, except if you were facing some sort of emergency or unexpected complaint.