I suppose that according to a classical theory (?), that wave could be perceived simultaneously by different people in different places around the source.

That is indeed what the classical theory says. It is wrong. This is where the assumption that domain of the amplitude is continuous is a bad approximation.

So how would Many Worlds work in this case?

Quantum amplitude flows into separate configurations. For each detector, there is a configuration such that that detector was the only one to detect the photons. There are also configurations where no detector detected it. So, if in some configuration, a detector detects the photon, and goes to check on another detector, it will find that the other detector has not detected the photon, not because some instantaneous space spanning signal collapsed the wave function, but because in that configuration the photon did not go that way.