There is no deadline and sometimes the family feel that the executor is moving too slowly. If so, and if you have a copy of the Will, you are entitled to start the proceedings with the probate court to get the ball rolling. You should talk with a probate attorney in your area if you plan to start a probate case.
There is no set time. However, an "executor" does not have any legal power to act on the estate's behalf until he or she files the will or at least a petition in probate and receives Court Approval of his/her status as personal representative. Regardless, probate may not be even necessary if there are no assets to probate. I have been hired to probate estates (and sometimes reopen the probate of an estate) years after a person dies when missing and hidden assets are discovered (the oldest issue I was consulted on was a person who died in the 1930s with property interests out of state that do not expire--I had to look back several generations of a family).
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There is a time limit, but it is vague. MCL 700.2516 states:
A custodian of a will or codicil or person having possession or care of a will or codicil shall forward it to the court having jurisdiction with reasonable promptness after the death of the testator either by delivering it personally or by sending it properly addressed by registered mail. A person who neglects to perform this duty without reasonable cause is liable for damages that are sustained by the neglect. A person who willfully refuses or fails to deliver a will or codicil after being ordered by the court in a proceeding brought for the purpose of compelling delivery is guilty of contempt of court and subject to the penalty for contempt.
You should talk to a lawyer and may want to have him/her send a letter to the person with custody of the Will to force their hand. Feel free to call me if you would like a consultation.
This answer is intended to provide legal information, not legal advice. Legal advice should be provided by licensed attorney only after full disclosure of all facts. If you desire a no-obligation consultation to obtain legal advice, please contact me at 586-268-4463.