Minimum time is as follows: once the application is filed the clerk "posts" a notice of the probate and a hearing can be held on the Monday ten days following the posting of the notice. So, it is a very quick fuse. Contested cases can take years.
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This is a question better answered by the atty for the estate, if you are a distributee under the will.
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The "prove-up" hearing on an application to probate a Will and appoint an Independent Executor cannot occur sooner than the Monday following 10 days after the date the clerk posts the notice (which is usually the same day it is filed with the court). How soon after the expiration of that period a hearing actually occurs can vary depending on a lot of things: how busy the court's docket is; availability of the applicant; availability of the applicant's lawyer; availability of any witnesses (normally there aren't any witnesses beyond the applicant). But, if the probate involves a dependent administration (where there is no Will) or where there are any contested issues, then it could be several weeks or months longer.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.
I agree with the other responses, but I would add that, having the Will filed with the probate court does not actually guarantee that there will be an estate opened, at all. Most states require the Will to be filed with the court, following the testator's death. But if there are no probate assets, then probate will not be necessary and the Will would never be admitted to probate. If there were assets titled in the decedent's name alone, then probate will be necessary. If the assets were all joint or had beneficiaries designated, then those assets would bypass probate, (and the Will).
Your summary does not include enough information to determine whether there will be a probate in your situation or not.
*** LEGAL DISCLAIMER I am licensed to practice law in the State of Michigan and have offices in Wayne and Ingham Counties. My practice is focused in the areas of estate planning and probate administration. I am ethically required to state that the above answer does not create an attorney/client relationship. These responses should be considered general legal education and are intended to provide general information about the question asked. Frequently, the question does not include important facts that, if known, could significantly change the answer. Information provided on this site should not be used as a substitute for competent legal advice from a licensed attorney that practices in your state. The law changes frequently and varies from state to state. If I refer to your state's laws, you should not rely on what I say; I just did a quick Internet search and found something that looked relevant that I hoped you would find helpful. You should verify and confirm any information provided with an attorney licensed in your state.