Why did three boys voice Wart in The Sword in the Stone?

The budget is to blame:

Part of the problem stemmed from a simple lack of money. Despite the success of One Hundred and One Dalmatians, Disney did not want to risk that much money on the film, and severely trimmed the film’s budget. It shows. If, for instance, you’re watching the film in a casual sort of way and just happen to think, huh, that shot of Kay eating a big leg of meat looks suspiciously familiar, that’s because it is suspiciously familiar: it appears in the film three times. Other bits were traced from earlier animation work, setting an unfortunate precedent.

This impacted the voicing:

The budget cuts also show in the voicing. This is arguably the weakest vocal film of any of the Disney films done under Walt Disney’s direct supervision (we will encounter a couple of upcoming films that are worse), a particular disappointment after the spectacular voice work for Cruella de Vil in the previous films. Most notable: the voicing of the film’s supposed protagonist, Wart, who was voiced by three different child actors. Two were brothers, and sound somewhat similar; the third was not, and sounds distinctly different. Given that the film is about growing up, this problem might have been overcome—except that the director inexplicably chose to leave all three voices in for some scenes, drawing attention to the problem that they were not the same actor.