It is obvious that if Bowler destroys time then Wilbur cannot follow him. At the same time, if Bowler does not destroy time, then Wilbur knows that whatever Bowler did with the time machine has resulted in the world in which he lives, and he cannot know how it would otherwise have been. He has no motivation to "fix history".
Fortunately, that is not actually his motivation for pursuing Bowler. His immediate problem is that he left the garage door unlocked and someone in a bowler hat stole his dad's time machine, and he is trying to recover it before his dad discovers this and punishes him for it. Thus he will pursue Bowler, not because he hopes to fix history, but because he hopes to recover the machine.
How he tracks Bowler is a separate problem, but there are at least two ways it might be done. One is if time machines have equipment to trace disturbances in time caused by other time machines. It appears that Doris is able at least to detect such disturbances, and thus it seems likely that this feature would be included in early time machines. It becomes useless eventually, of course, because the more time machines move through history the more disturbances pass through the present--and indeed Bowler's trip might be entirely in the past by the time Wilbur pursues him. The alternative is that Wilbur heard Bowler exclaim that he was going to travel back and destroy Cornelius' life by taking that Memory Scanner from him, which would give Lewis a time and place to check.
Either of these solutions is made more plausible with one minor adjustment to our reconstruction. We know that Bowler stole the time machine, and that Wilbur pursued him. Yet it is unlikely that even Doris knew how to operate the machine the moment it was in their possession, and so they probably flew it out of the garage and took some time interpreting the controls. Since Wilbur saw them leave, he immediately gave chase. He might then have left the future before Bowler did. If he heard Bowler raving about his next stop, he easily might have engaged the time travel circuits and rushed back to our time to intercept him. This, though, leads to one of two odd anomalies.
If we assume that Wilbur heard Bowler declare his destination, and Wilbur managed to leave first, Wilbur arrives first, sequentially (whether he arrives earlier or later than Bowler is a separate issue), and finds that Bowler never arrived. This is because once Wilbur leaves for the past, all of history must be rewritten to the moment of his departure before it can continue, and thus before Bowler can also leave. Probably Wilbur will spend a few days in the past trying to track Bowler, then return to his own time. This in essence unlocks history so it can progress, and Bowler leaves for the past. This changes Wilbur's experience in the past, because now Bowler arrives and Wilbur attempts to recover the time machine, and more pointedly thwart Doris' now unfolding plan.
If, though, we assume that Wilbur is tracking Bowler's time trail, we have the oddity that Bowler must leave first sequentially, but Wilbur must leave first temporally. That is, if Bowler leaves at ten o'clock, then his trail runs from ten to nine to eight and back into the past, and by eleven there is nothing to follow, thus for Wilbur to find Bowler's trail he must find it before ten. Yet it does not yet exist at nine until Bowler leaves from ten; but Bowler does not stop at nine, but travels back about twenty-five years. Thus Bowler leaves at ten and arrives in the past, and history rewrites itself up to nine o'clock, at which time Wilbur is looking for Bowler's temporal trail, finds it traveling backwards from ten and follows it into the past. Wilbur's anomaly must then resolve in order for Bowler's to do so.
If this were all of it, it would resolve fairly simply. However, Wilbur does something that proves fatal to the story from a time travel perspective. We can excuse him for this because it was a snap decision under duress, but at this point there is no saving the movie. That will be covered next time.