The phrase refers to any informal method of transmitting information or rumors swiftly and unofficially by word of mouth or other means, such as runners, drum codes, or smoke signals. "The bush" refers to extremely rural, even primitive, conditions.
It originated in Australia as slang for any chain of communication by which criminals, known as bushrangers, let one another know about the movements of the official police.
The narrator uses the phrase to refer to the rumor and gossip that surrounds Tomas after his essay on Oedipus and Communist collaborators is published, for which he eventually loses his job. The narrator mentions "bush telegraph" again a little later (section 8 of Part Five, page 197), after Tomas has become a window washer, and the word quickly gets around that he's really a doctor.