Yes, things can go smoother with the Medicare/Medicaid process. Ours took about 6 weeks because it took 2 extra weeks for the inspector to come out, no Texas Medicaid number allowed until after you get the Medicare number, then about two months to get most of the HMOs completely registered. But when starting a business, you should always try to expect the worse case (or near worse, as getting denied the number would be the worst case). This is why you should expect a 3-4 month no revenue period.
If you aren't running during this time and just waiting for the paperwork, this isn't too bad. BillingSpecialist may know the answer in regards to Medicare still back dating your application so your number is effective on the application day instead of the approval day as I know they were making some changes last year.
I advise to plan for everything to take twice as long as you originally thought and cost twice as much.