I was chatting to a senior US banker on this subject last year, and he advised me that unlike during the depression, there is deposit insurance now in US banks, I believe the deposit insurance is higher in the UK than in the US, so probably worth checking out individual circumstances before panicing. Being a common shareholder when a business goes belly up or bought for a song is a different scenario than being a depositor in it.
One of the reasons why there was a depression was because people took money outof their bank account out of fear, which meant there nothing left over to lend to business... we'd all rather that didnt happen- anywhere