The smallest possible interval is
just over 1 second.
The other answers correctly note that the nearest palindromic times are
9:59:59 and 10:00:01
But they incorrectly conclude that
2 seconds must elapse between them. This is incorrect. If we look at the clock in the last possible moment that it reads 9:59:59, we just have to wait 1 full second while the clock reads 10:00:00 before it reads 10:00:01. It is possible to see two palindromic times on the clock while looking at it for just over 1 second.