There are two sets of three pips correctly numbered with respect to each other 1, 4, 9 and 2, 5, 12.
There are four pips, numbered 6, 7, 8, 10 that do not relate to any other pip.
There is only one set of two related pips 3 and 11.

The traveler was told that he could tell the time if he knew how many pips were correctly numbered.
The pips numbered 3 and 11 (a unique set) must be correct otherwise the traveler would have had a number of choices (2 sets of 3 related numbers, and 4 individual numbers)..

Counting counter-clockwise from pip number 11 shows the hour hand pointing to what would normally be 9:00 o'clock, confusingly re-numbered as 2.00 o'clock. The traveler was also told that 12:00 o'clock was no longer at the top of the dial face, so he now knew that the dial face had been rotated 90° counter clockwise.

The traveler met the passer-by in the morning, hence the time is 9.00am.