There are a few things here
Having a date when probate was granted when there was no will needs some clarification.
You certainly need an update on progress from the solicitor.
Once letters of administration have been granted, the estate can be distributed. In your situation, that probably means that the house needs to be put on the market and actually, you know, a buyer found and actually sold. Has an estate agent been instructed ? By whom?
I would agree with PP suggestion that you take control of that process yourself. Then instruct a solicitor as usual for the sale.
As an executor, It's not usual to distribute the estate until you know how much there is, for example what legal etc fees will be required to sell the house.
If there are only the two of you to inherit, though, once you have letters of administration, I see no reason why you shouldn't share the liquid assets now and then split the proceeds of the house sale when they become available
The only risk would be if the costs of selling the house are greater than any equity available from the house sale.
Inform yourself about what needs to be done at each stage. I'm not sure the solicitors are doing a good job. I have done several estates, always need a solicitor at some point but the process itself is essentially administrative.
If you can work out what needs to be done you will be in a better position to chase up the solicitor. We are still waiting for probate to be granted on one estate, still the lovely solicitors reply's to any question within 24hrs. If it will take longer to get an answer, they send a holding email.
All the best.