The solution can be found without pencil and paper!
Let D be the number of kilometers between Town A and Town B. When they first meet, the sum of the distances they have traveled is D, since they have started D kilometers apart and have closed the gap between them. This is true no matter where they meet! Since it is 10:00, they travel a total of D kilometers in two hours.
By similar reasoning, they have traveled another 2D kilometers by the time they meet again on their return trips, no matter where they meet. They have therefore been on the road another four hours and taken a half hour for lunch. It must be 2:30 p.m.
Notes: The number of minutes since Bob ate is irrelevant, except that it establishes that they are both on their return trips when they meet the second time. We got some , beautiful, elaborate algebraic solutions from some strong solvers that used the 45 minutes since Bob's meal to find the ratio of the two riders' speeds and the exact locations of the two points where they meet, and then used these to derive the time of day. In one case, a solver showed great algebraic insight and prowess, but due to an inconspicuous error buried in the derivation, concluded that the time of day was irrational!
The problem is a great prank that seems to work best on strong mathematicians.