How exactly would they interfere?
In textbook/Copenhagen interpretation, we say that when the electron passes through the slits the wavefunction has not yet been measured. But with the detector in place, you collapse the wavefunction to an eigenvector of position. By the uncertainty principle, the momentum uncertainty is now very high, so the electron shoots off in a random direction and cannot be expected to follow any pattern.
In MWI, we start with the interference pattern. But adding the detector does not "collapse" anything - instead it introduces lots of degrees of freedom. The particle passes through both the left and right slits, but the degrees of freedom ensure there is no fixed relative phase delta between these possibilities. The detector turns waves into "static noise", and so any interference pattern is lost. Both options are realized, but cannot detect the other possibility: distinct Worlds.
The key point is that MWI has no special role for measurement. The left-slit and right-slit worlds interfere, but with the detector in place the pattern is destroyed: the interference is uncoordinated and averages to zero, so undetectable. Whereas in textbook/Cophenhagen QM we say that the measurement results in a single outcome.
I had a discussion with someone on HN about this a while ago and realised that "world" or "split" isn't necessarily synonymous with "superposition". Rather, I believe that a split occurs when the superposition entangles with the environment and causes sufficient decoherence that there is negligible probability of (measurable) interference.
In principle you could imagine a thought experiment where you had a super-powered quantum machine which finely could control the quantum state of a large, isolated room. In that situation you could imagine someone in the room conducting an electron spin measurement and looking at the outcome, before the machine enacts a reversal of the room's quantum wave-function, thus causing the two copies of the person to interfere. If we ever reach that level of technology, it will be fascinating to see how the interpretation debate progresses.