I believe in Many-Worlds, the update step falls out of decoherence "for free." Because you are also in superposition after the measurement, the "you have measured 1" part of your wavefunction of course measures amplitude 1. That doesn't give you the Born rule though - I believe there's some attempts to get it out of game theory [1], where they show that given sufficiently long timespans, "almost every" worldline has a history with quantum measurements whose distribution matches the Born probabilities.

[1] https://arxiv.org/abs/0906.2718