Just been through this process.
First you need to get probate , the value of the shares for Probate is the value on the date of death.
Once you have obtained probate , then you have two options. Either transfer the cerificated shares into the Name of the Estate once that is done you can sell the shares. You will need to set up a Bank Account in the name of the Estate, most of the high street banks will do this. You will need to send the original share certificates plus the probate certificate to the Share Registrar. Most of the Registrars actual have a dealing service for the main UK Shares, if you simply want to sell the Shares ( if the shares are sold within one year of death and for lower than the probate value then you can apply to get any Inheritance tax back. If sold for more the Estate may have a CGT liability depending on the actual gain – Estates get an independent CGT allowance in their own right, as well as an Income Tax allowance.) The second option is to Transfer the shares directly into the Name of the Benificiaries if they want you to, for future CGT calculations the carrying/transfer value will be the value used in the IHT calculation. The Benificiaries are then free to sell the Shares or hold them.
As an Executor , you have a duty of care – you are not theere to speculate on the market.
DM me if you want any other information. It is actually very straight forward.