BIP did not close in 1939.They had been renamed Associated British Picture Corporation. Although their studios at Elstree were closed at the commencement of war, they continued to make movies at Welwyn Garden City. Consequently, their movies could not have acquired the status of public domain.
As rights to the BIP studio movies became public domain soon after they closed in 1939, this movie found its way into the inexpensive early 1950s American movie package for television sold by "Atlas Films". Atlas scooped up piles of public domain movies like this and chopped off original titles, replacing them with minimal information new titles, often changing the name of the movie, in this case, "The Doctor's Secret".