My prediction: First, OSHA has to actually draft and publish the proposed new rules - which don’t yet exist. And they have to provide time for review and implementation.
Sometime shortly after publication you’ll have a jurisdiction-shopped district court judge brought a case for injunction against the rule (almost certainly in the 5th Circuit states), chosen knowing he’ll grant it. Then the 5th Circuit will uphold the injunction regardless of precedent or the law. Then finally the Supreme Court will get its say - but barring crystal clear precedent, I’m guessing they’ll at a minimum uphold the injunction, and at worst they may use the opportunity to basically gut OSHA if the Trump “Justices” are having a bad hair day.
What Biden should have done instead (or in addition) is mandate vaccinations for all age 12+ travelers on planes and trains, which the federal govt has much more direct and clear broad interstate authority over, and could likely be done on shorter notice.