Op my mum died just over two years ago, straighforward will, DB and I executors and beneficiaries and, like your uncle, a house to sell and some cash.
It was very straightforward. We both stayed as executors but I did most of work. The probate forms are very straightforward if you are below the IHT limit. It should be very easy for you or your dad/uncle to help fill the forms in, then take the aunt and uncle to a solicitor for signing them.
The oath can also be done at a probate office, we did ours at the London probate office as it was easier for my brother, I just took train down there.
For house sale, you need a valuation for probate. Like you, we wanted mums house on market ASAP so got 3 estate agents to value it and took average for the form. It went on market before we applied for probate, sold instantly, and we then had probate certificate back during the 2.5 mths the sale was progressing.
You also need a value for other goods and valuables for probate, but since most of mums went to a charity shop, I just estimated a value for the bits we might sell. Think it was 1k.
Accounts need to be kept for probate, but can just be a simple spreadsheet listing assets and any debts eg oustanding care home fees, final house bills, any tax owed etc. I would be more than willing to send you ours with all figures removed, as an example.
Once probate certificate is received, you can send it to banks and they will release the cash. Whilst you can set up dedicated executor accounts, that would have been hard for DB and I, not being local, so we used a current account that I had retained simply to put cheques into. I moved all except for £1 out of it and gave the banks the sort code/account no in a letter when we sent probate certificate off.
Hope this helps, I wouldn't involve a solicitor if you dont have to.
My bachelor cousin died 3 yrs ago leaving his estate to cousins children and a solicitor as executor. It took much longer than mums estate to sort, and we then had discussions with solicitor, for us to take control of DDs money. She was 16 and the only beneficiary under 18. Solicitor suggested we take control, but then took 8mths to sort the paperwork and do the money transfers! I came very very close to reporting them, I got so frustrated with lack of communication and action.