We are still waiting for your rework schedule. Please provide us ASAP as the shipment is overdue.
Of the second part, I would ask, please provide what? I fell there is a need for "this" in there: "Please provide this ASAP" (that it is to be provided to us is clearly evident/implied in the context).
We are still checking with the factory for the schedule and hope to reply you shortly, please be patience.
Either "reply to you shortly" or simply "reply shortly" would be better. And it's please be patient (the adjective), not patience (the noun).
Student or Learner