While there were no passengers with the last name Taylor outside of first class passenger Elmer Zebley Taylor (1864-1949) and his wife Juliet Cummins Taylor (1862-1927) as correctly mentioned by Seamas there were a number of Taylors on that were part of the ships crew. Here is a brief list of all of them:
Third class steward Bernard Cuthbert Taylor (1889-1912)
Able bodied seaman Charles William Frederick Taylor (1871-1912)
Fireman George/James Taylor (1888-1931)
Fireman John Henry Taylor (1862-1912)
Fireman J/T. Taylor (1870-1912)
Turkish Bath attendant Leonard Taylor (1893-1912)
Fireman William Henry Taylor (1883-1914)
First class saloon steward William John Taylor (1881-1912)
Violinist Percy Cornelius Taylor (1872-1912) (the instrument he played is debatable).
The plan of the brothers Taylor are unclear if they were intending to travel as passengers, or as crewmembers in the article if the story is indeed accurate. As far as documentation goes, the only person that currently is listed as being born in Leicester is the master-of-arms of the RMS
Carpathia at the time (a man named John Brown). Personally, I can't deny I'm a bit skeptical of the story as far as documentation goes, however I do look forward to see more information coming from your side. Keep in mind however, that telling and repeating a story isn't always true. Just look at Helen Kramer her story. The Titanic her story has countless claims like this, and most of them aren't true sadly. If every story was truthful the ship would have been fuller then her passenger's certificate would allow.
I hope this might could offer some leads in the right direction.