Chesterton didn't write it.its credited to jude Tindall,but plot and method of killing is very similar to green for danger .1946 film with Alistair sim.from a novel by chritianna brand.
I was not paying close enough attention to be certain, my companion viewer thought the episode anticipated the coming of NHS (5th July 1948) whereas I thought it was written as if NHS had already started.
The previous episode was about a jewel sent by the Pope to the Queen for her Coronation, which was, of course, in 1953. If this episode took place around that time, the NHS would be about 5 years old (started July 1948).
The story borrowed both from "Green for Danger" as previously mentioned, as well as from "The Nursing Home Murder", an Inspector Alleyn novel.
Borrowed from? It was outright plagiarism of Green for Danger. I haven't seen The Nursing home Murders, but surely if they did not credit Christianna Brands work, they could be sued. Even if it is in the public domain, it was badly done by the writers. Very unethical not to credit. I did not see whether they did or not, to be truthful.