How can we find out where mother had shares?

Our mother, now passed away made it known that she has shares, mainly bought in the 80s when Water, telecom etc were sold off.

However, we have never managed to find a list or paperwork with account numbers or similar.

Is there a central register of sorts where we an track down the shares she held so that we can wind up her estate? (Such as it was after the care home took most of the money from selling her house .)
(No solicitors involved. Son and daughter sole beneficiaries.)

Thanks in anticipation