Teaser 2881: A month of meetings

From The Sunday Times, 10th December 2017 [link]

In one particular month this year I had one-day meetings in each of Geneva, London, Rome, Tallinn, Venice and York. For any two of these cities their names had at least one letter in common precisely when the days of their meetings (1st, 2nd, 3rd …) had no common factor larger than one. No two meetings were on the same day of the week (e.g., no two meetings were on Wednesdays). The Geneva meeting was the first and the London meeting was the last, the London meeting being on a Friday.

What was the date of the Tallinn meeting (month and day)?

The original puzzle text used the spelling “Tallin”. In the text above I’ve used the more usual spelling, but the puzzle works with either.