There is an absolute wealth of information about WW2-era sonar and underwater radio in
this document from the US Navy Technical Mission to Japan. Although the focus is on Japanese technology and tactics, there is information about German and early-war British equipment too, the latter having been captured by the Japs at Singapore. There is also data of considerable general significance from Japanese research of the time.
Of particular note is that the Germans built a VLF transmitter in Japan (complementing their own transmitter in Germany) to assist communications between the two countries when shortwave transmissions were unreliable, and that this transmitter was used operationally for the submarine components of the Pearl Harbour attack. The German transmitter was reported to be received underwater off the South American coast.
The wavelength of over 7km means that a VLF transmitter requires an absolutely enormous antenna, which would be hard to build without official notice being taken. However, I note that Sweden already had extensive electrified railways by the late 1930s, and at least in theory, the overhead power wires could have been adapted into ersatz VLF antennas; it is usual to divide such equipment at roughly 10-15-mile intervals in order to distribute the traction load over the three phases of the power grid. A ten-mile section would make a great horizontal dipole on a 32km wavelength - about 10 kHz!
At such low frequencies it is necessary to use a very tight bandwidth (on both transmission and receiver), which necessitates a comparatively slow signalling rate. The modern applications of submerged VLF employ such low signalling rates (to ensure correct reception) that often all that can be sent is a single codeword, instructing the submarine to periscope depth to receive a longer message, or triggering some order previously detailed.