I'm no financial professional either @WomensLandArmy but have recently been in the sad position of having to sell shares belonging to my late husband.
I had sold some shares of my own a few weeks before he died, and paid the sum of £25 for the service with an on line company.
Simple and straightforward.
Sadly, they don't deal with sale of deceased shareholder's shares so I went direct to the administrators of the shares (just Google who administers x company's shares and it should pop up).
I found that it wasn't as simple as selling your own shares - nowhere near as simple.
The shareholder's death had to be formally registered in the first instance and in order to do this, they wanted to see the Grant of Probate document - possibly other paperwork too but I can't remember exactly (I've sent a LOT of paperwork to a lot of people recently!).
Once that was done, and I had been notified of the formal registration , then a form was submitted to sell the shares.
There were a lot of instructions regarding the way the process had to be completed and I was advised that not following it would incur extra costs to me eg. I couldn't send the Grant of Probate and the instructions for sale all at once otherwise I'd be charged £60 admin costs over and above the fees for handling the sale.
I was charged £70 in total for the sale of the shares - and the value of them was nowhere near your 40K value.
If there are paper share certificates involved, I would strongly advise you get copies or photographs of these before you send them anywhere to sell, as I was also charged a £60 "lost certificate fee" but was able to challenge this as I had photographs of them and could give the company all the details regarding certificate numbers etc etc.
I'm currently awaiting reimbursement of that fee.
Good luck with it all.
It's not simple but is doable providing you read the small print.