UK national health insurance ONLY covers people WHILST they are in UK or a jurisdiction with whom UK has a reciprocal arrangement. WithIn UK of course there is no issue of rejection for reasons of poor health etc. But the view is that it is not up to the tax funded scheme to pay for people's health matters whilst they are swanning around the world on holiday (and if they go abroad on business then it is up to their employer to cover them).
The UK system is also far from " free" - many people will pay 12% of their salary (those who are unemployed or on low incomes get it for "nothing" - hence one of the concerns regarding immigration). This sum however is also paying towards state pension etc (although Ordinary taxes do this as well).
The BIG difference seems to be the UK practice of NOT covering illness whilst abroad from what is a "National" health scheme.

The tour company just cancelled my upcoming trip across Turkey, which makes ALL their WHSes "too dangerous to visit."

From which country is the tour company?

Solivagant
The UK, I believe. G Adventures has offices there, Canada and Australia.

It is primarily a Canadian company,
Founded by Bruce Poon Tip in 1980.
They "guarantee" departures once you have booked, apart from "safety reasons"!!

It looks as if the Bamiyan/Jam area isn't quite ready for tourists yet!! But you can buy the experience of getting attacked by the Taliban for a mere £2500 by going for the "shortened" tour (plus flights but you could possibly save on them by only buying a one way ticket??). The attack took place a bit west of Jam (which according to the schedule they would have just visited!)
https://www.theguardian.com/world/2016/aug/04/foreign-tourists-attacked-in-western-af ghanistan-says-officials

I see that the same travel company also claimed (it isn't absolutely clear how up to date this is) currently to be operating tours in Iraq south of Baghdad if anyone wants to pick up this year's new inscription.

Last September at least they made it unto the Marshes:
https://www.facebook.com/HinterlandTravel/posts/851627381611802

Tourism in Afghanistan... Sure fire way to get a Darwin Awards.