Not exactly. The Orthodox Jews run their own service in a lot of places. https://en.wikipedia.org/wiki/Hatzalah

I'm not familiar with situation in the UK, but in other countries there are NGOs that try to complement the national healthcare services, ranging from financing ambulances to at home care. They are often tied to religious movements.

At a quick glance, Hatzola Northwest seems to provide services for free to everyone in their catchment area, so any segregation is geographic in origin.

Nope, it is an adjunct funded by the Jewish community to avoid accidental things not approved of (can't list what as I can't recall).