To answer this, I would turn to contemporary guidebooks. Basically you have the routes through Russia or through Constantinople. Baedeker’s Russia of 1914 suggests a ferry from Odessa to Batumi, or rail from Rostov to Baku. The 1893 John Murray Russia guidebook suggests St Petersburg-Rostov-Vladikavkaz-Baku by rail or Constantinople-Batumi by sea. The table of contents in Murray suggests a number of routes from London to St Petersburg or Odessa. The Murray Constantinople guidebook (1871) outlines routes from London to there.

You might also have a look at John Murray’s Turkey (1895) which also covers Transcaucasia and Persia.