My searches have brought up a source that may have been influential, between the 1940 example she gives of Thurtle and the dubious 1972 journal extract: Pierre Daninos's Les Carnets du Major Thompson (1954), which was made into a film starring Jack Buchanan, and which (in its English translation) has the Major's wife Ursula being advised by her mother to "just close your eyes and think of England!"

I wouldn't be surprised if that book gave the phrase quite a boost, at least. But Major Thompson is a fictional version of the typical Englishman, and one invented by a Frenchman, no less - so certainly a case of "He would say that, wouldn't he?"