A view from the inside
A few points in reply to some of the comments made on this story.
1) No - we don't use thermal paper. For the love of God, it's 2018 !
2) Many of the "fax" machines are indeed part of a multi-function copier which in it's default configuration under the OGC (Office of Government Commerce) contract, has the fax option installed as standard.
3) Many departments have an analogue line for fallback, in case the VOIP system fails (and it can), so even if you rid the NHS of fax machines, there will still be a large number of analogue phone lines in use.
4) Fax isn't used anywhere near as much as it used to be - it is on the decline, but it's taking some considerable time for each use case to be resolved with other technologies.
5) Fax as a protocol is far more capable than you'd think. Unfortunately, the full possibilities never really took off much outside of Japan. Colour fax was actually a thing there !