Rhodes, although a Greek island, was under Italian occupation from 1912 until the end of the second world war. Before that, it had been part of the Ottoman Empire from 1522 until 1912, when the Italians helped the Greeks to throw out the Turks.
This explains why, in this episode, the police are Italian and not Greek.
The shots of Poirot's Whitehaven Mansions building were taken at Florin Court in Charterhouse Square, near Holborn, London EC1.
At the start, Hastings is listening to the radio coverage of a cricket test match, and the commentator refers to the Sri Lankan team - but Ceylon only changed its name to Sri Lanka in 1972, and this episode is set in the 1930s.
At the end of the scene where Poirot and Japp are examining the murdered girl's body, there is a close-up on the body. As this happens, you can clearly see one of its fingers move.