The following is a combination 'extract/paraphrase/own' largely from "A Polish Question: Monarchial sentiments in Poland 1918-39" by Michael Nash, Royalty Digest #167, May 2005.
In October 1928 The International Herald Tribune quoted the Prime Minister of Poland as saying "Poland needs a monarchy...if the country is to prosper."
Poland was a kingdom until its dismemberment in 1795, and during the period of elective monarchy sovereigns had been chosen from outside the country, so the precedents for a monarchical system and an outsider King were already established
The leading (unofficial) contender was the Duke of Kent, son of the British King, handsome, socially at ease, and married to the glamorous Marina, who came with impeccable Royal and Imperial connections. In August 1937 they went to Poland to see for themselves, and in some quarters at least were rapturously received. Though not an offical state visit, in many ways it resembled one. They visited several towns, where the Duke gave speeches praising the Polish people.
From then on, the Duke of Kent studied everything about Poland and Polish affairs, eagerly educating himself for his future role. According to a British intelligence writer, the idea of making Kent the King of Poland originated with none other than Lord Louis Mountbatten. [The prospect of Mountbatten's younger daughter marrying the son of the Duke to become Crown Princess and eventual Queen of Poland is not canvassed.]
The Duke made many trips to Germany and Central Europe during 1938 and 1939, usually interprteted as social visits to relatives (Prince Paul in Yugoslavia, the Toerring-Jettenbachs in Munich, the Hesse-Cassels in Hesse etc). Whether they were anything more is purely speculative, as the adventure, if it ever was, came to a sudden end in August 1942 when the Duke was killed in an air crash in Scotland while on active service.
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On another note and another "what might have been", the author states that the widowed Princess Marina was asked to become Queen of Norway by the also-widowed King Olav V after he became King in 1957. The exact timing of his overtures to Marina and to the Queen Mother, and whether they overlapped, is unknown.