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It's sometimes possible to get the hospital to confirm that you are covered through some national contact point and to have the relevant institution back in your country of residence (the NHS trust in that case?) fax a written confirmation. But a friend of mine in a similar situation did have had to pay bills upfront and claim the money later.
So the conclusion seems to be that your mileage may vary and the exact combination of countries and healthcare providers may matter. Do ask if they can somehow get a confirmation that you are in fact properly covered by the system. Some useful contact numbers are available through the official app.
Importantly, coverage is based on residency and/or affiliation to some specific healthcare system, citizenship does not matter as such. Similarly, the fees (including things like copays) depend on the country where you are getting care and not on your country of citizenship or residence.