Find shares in dead relatives name?

Last September, my father died. We sorted out all the finances and divided it up equally between me and my siblings. However, my uncle contacted me the other day and provided me with a letter from a company saying that dividends for ~£100ish pounds could be claimed, so there's some shares he owned that are worth a bit of money that I'm currrently in the process of transferring ownership to myself so I can divide them up.

However, we know my father used to invest in various different companies and I wouldn't be surprised if there were more shares he owns that have yet to be discovered. However, my father had mental health issues for the last decade or so, so combining that with his non-existent filing system as well as quite a few addresses over the last 10 years means that trying to track down what companies he might have had shares with is a nigh on impossible task.

Any suggestions on how I might be able to find out?