We bought a probate house and as I understood it, one the seller solicitors had sent off the probate paperwork there was no way for them to know how long that was going to take to come back.
I appreciate you are saying that they have not confirmed they have sent it, but assuming they have I think there was very much a sit and wait period.
The situation we were in though was we had a holiday booked and wanted the purchase to go through after that to save paying for a holiday place, the rented house, and the house we were buying all at the same time so were hopeful this would hold things up. In this time the solicitors I think drew up drafts of everything in anticipation of the probate being granted and we did end up having to buy the place then go on holiday the next day.
I think the probate thing took about 6 weeks to 2 months to come through, on a simple one beneficiary inheritance, and once it did things then moved fast. Like I say there was no way of getting any indication of how far off the probate was once the paper work went in, I think I was told it could be another month or 2 one day, then the next it was all done.