How long is a piece of string? If the estate of non-taxable and assets fairly straightforward, no reason why they can't apply for it within 3-4 weeks of death - probate registries can deal with applications fairly quickly and normally turn them round within 2 weeks. Problems arise if the estate is large and funds have to be raised to pay the IHT or if there is difficulty in tracing or valuing the assets.
This is one of those 'how long is a piece of string?' questions. It depends upon how complex the estate is and how efficient the solicitor is.
3 to 6 months is quite usual. 2 years (or longer) isn't unknown. However, when my father died, I sought probate myself. (Unless the estate is particularly complex, there's absolutely no reason to use a solicitor. It's normally a very simple task). I had everything completed in under a month.
The thing that takes the time is not getting probate - that's quick - but gathering the information necessary to apply for it. Solicitors don't tend to rush this - nor do the firms responding to their queries.
Nor do they rush to sort things out once they have probate.
As Chris says - if you have the time and the ability, DIY.