I just thought of something else.
In the original law article to which they refer in the letter, they following phrase is being used:
Mütekabiliyet esasi da dikkate alinmak şartiyla, oturma izni almiş yabanci ülke vatandaşlarindan yabanci bir ülke mevzuati kapsaminda sigortali olmayan kişiler.
Translated:
With the reciprocity principle taken into consideration, foreigners with residence permits who are not insured under the legislation of a foreign country.
Maybe UK citizens might get away with the reciprocity principle.
All Dutch (living in the Netherlands) have to pay monthly for a mandatory health insurance.
For Turks living in the Netherlands on a residence permit, this insurance is mandatory as well and they have to pay on a monthly basis.
UK citizens don't pay for NHS and according to a quick Google, it seems that foreigners living with a residence permit for longer than 1 year in the UK don't have to pay for the NHS either.
This would mean that the reciprocity principle does not apply to UK citizens (whereas it would apply to Dutch).
Or do I understand the situation incorrectly?