Mind you it has been 3.5 years, but we got both of ours quite quickly. Under 6 weeks I would say.
It may also vary by the county. We are in Wayne County and they do the name change at the court appointment and then send a copy of it to Vital Records requesting that the Delayed Registration of Foreign Birth be issued. So depending on how quick the court is to mail the request would affect how long it took.
So not much concrete information, but for what it is worth!
You might call the court or Vital Records in Lansing and see if they can give you the status.
Good luck!