If I'm making the rules, I'd start on the disposal for the very reason Aremred states. Technically, you could justify starting the clock as soon as the timeout is over, but I'd compromise and do it when the ball becomes live.
I honestly don't normally pay that much attention, though, and just let the clock operator do her thing. It's either a mom, dad, or sister who doesn't need me piling on to the stress, or it's someone working the entire tournament for a pizza and tacos and they've got their system in place already.
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