Ahhhh! I can actually answer this. Yay!

The ww2 era British military used a ton of different methods to communicate, but classified telegrams specifically were sent over both dedicated military lines (which were fairly obvious as they were the brand new shiny lines freshly laid by the military) and civilian lines (in the hopes that maybe the traffic would go unnoticed). Also radio was used for bits of it wherever actual physical lines were unavailable; a single communication message could travel via radio to a spot with lines then back over radio, then back over lines, until it got where it was addressed.

(You should check out the history of war pigeons sometime- they used carrier pigeons for some communications where they couldn't manage radio or wires.)

Anyways, that's the physical layer. The second layer is content.

In some communications you'd have obfuscation- you'd want to send a message so that the enemy never even knew a message was sent. This would be your public communications, so you'd use the old classic stuff like calling a bomb a football and discussing armies like sports teams. Both sides would have set references they'd understand so that the conversation would be understood.

In almost all communications though, you'd have some level of encryption. While most governments have used encryption in their sensitive communications throughout history, ww2 was the first time mechanized encryption was really a thing. The history of competing encryptions between the axis and allies is awesome. It's also absolutely massive and covered in huge, excruciating detail in a ton of other places, so I'll leave it alone. Check out Bletchley park sometime though, it ties into the basic history of the origins of the computer.

So this was the beginnings of the encryption revolution, which means everyone wasn't all on the same level- you had this crazy array of encryption methods in use, everything from simple substitution cyphers to one-time pads to book cyphers to basic mechanical stuff to the more advanced stuff in use by commanding officers on the front lines trying to talk to the senior staff in the rear.

And yeah, stuff got captured and decrypted all the time, it was a big deal. Folks died a lot. It was part of the war. :)

Simon Sing: The Code Book

Christopher Sterling: Military Communications: from ancient times to the 21st century