Yes it is a legal requirement to display company registration details on a UK based website. It's also good practise - it'll will reassure the company's customers that it's a serious organisation.
Out-Law site contains the relevant information.
The name, geographic address and email address of the service provider. The name of the organisation with which the customer is contracting must be given. This might differ from the trading name. Any such difference should be explained – e.g. "XYZ.com is the trading name of XYZ Enterprises Limited."
And according to The Register from January 2007:
Every company should list its company registration number, place of registration, and registered office address on its website as a result of an update to the legislation of 1985. The information, which must be in legible characters, should also appear on order forms and in emails.
This is part of the Companies Act and failure to do so will result in a fine.