So, your bigger problem is not your program (though that may be a problem) but your temporary license (or training license, or however your state deals with it). You cannot start work as a physician in any capacity without a medical degree. It's flat out illegal. That said, very few programs actually start treating patients before June 16 - you might have *orientation* then, but sitting around listening to HR lecture about benefits or some administrator blather on about duty hours/fatigue mitigation doesn't require a medical degree.
How long it takes from the day you send them the degree until you're legally allowed to start work is probably highly state dependent. Most of the time, the training license is super streamlined (or it may even be automatic under the auspices of your attendings/institutions license), but you absolutely need your diploma before you can see a single patient as a resident.