This answer explains how to activate the .uk WHOIS privacy option in the 123-reg control panel.
Please Note: .uk WHOIS privacy is only available to private individuals.
How to opt out
In the Domain names section, select the relevant domain name in the drop-down list and click on the Manage button.
Click the WHOIS Opt Out link.
Tick the I Would Like To Opt Out box.
Select your registrant type from the drop down box.
Click on the Go button.
Please note: To opt out, you (the registrant) must be and remain a 'consumer', i.e. an individual who has registered and is using the domain name for a purpose unconnected with any business, trade or profession.