The Financial Services Compensation Scheme (FSCS) is the UK's statutory fund of last resort. At the moment, the threshold for deposits is £35,000 ($62,580 today) but is expected to rise soon (autumn 08 or early 09) to £50,000 ($89,400). The threshold for investments is £48,000 at present.
The compensation limit of applies to each depositor for the total of their deposits with an organisation, regardless of how many accounts they hold.
If people hold multiple accounts in banks that are part of a larger group, if each of the banks is separately authorised by the Financial Services Authority, FSCS would pay compensation up to the limit, per authorised institution.
If each of the banks is not separately authorised but is covered by the parent company's authorisation, FSCS would pay compensation up to the limit once, irrespective of how many different institutions a person held accounts with.
So for 100% peace of mind, it might be worth thinking about that.
However, Great Britain is not a third world country, its GDP approached 3 trillion dollars in 2007. I would not be worrying too much.
http://www.fscs.org.uk/consumer/For comparison, US depositors have FDIC protection of up to $100,000 in the event of a bank failure. I understand it is common practice to split larger sums across different accounts to ensure protection.