I had the same problem 4 weeks ago, for 3 times at my 2010 Roadster S3 (with start button, not sure if this makes a difference). The former owner reported it in 2 occasions. Full battery, all seems ok, all control lights on, but no click / start.
Problem was that the starter was "hanging, had not gone back to zero position. It helped in all situations to knock against the starter with a hammer or similar (under right bonnet, cylinder with red mark for plus / jump-start), not strongly, just to support the mechanism to go in original position.
This happened when I was driving for a longer time in very warm conditions. Since then not a problem any more...
I had the same problem 36 years ago with my first VW Golf series 2, where a hammer was my best friend… :-)
