It takes the same as if there is no will.
Currently, in Dublin, papers lodged by solicitors for Probate ( testate or intestate) are taking 2-3 months to come through. But it can take weeks, months and even years after the death of the person before the family come around to dealing with it.
Is your question how long for Probate after papers are lodged? By solicitor? Personal application?
Or how long does it take most families to get around to it? or how long after full instructions are given to the solicitor?
mf