Who's doing the work? Are you seeking probate in person or doing it through a solicitor? My father died on 16th October and I got the grant of probate on 25th November, so that's about 7 weeks from his death. I probably didn't start the process immediately after he died though, so perhaps 6 weeks might be a better figure.
However:
(a) that was over twenty years ago (although, as far as I know, nothing much has changed in the way that the Probate Registry works) ; and
(b) I did all of the paperwork myself. (For example, I actually visited the office of his pension provider, with his will, a copy of his death certificate and proof of my identity in my hand). I suspect that a solicitor would have taken a great deal longer over the task.