If it's any kind of business registered with Companies House (plc, ltd, charity, etc), the details of the directors must be publicly available - by law. Not just names, but registered company address too (although that can be a legal representative, such as a solicitors). Anyone can find out such info from Companies House (for a fee).
Be wary of registering domain names under an alias, you might find you don't legally own it! You can, however, register a domain name in the name of the owning company. The other details, technical & admin contacts, could be hidden or obfuscated.
And bear in mind that the rules are relatively strict when it comes to online trading - if you've got site representing a UK business you need to follow guidelines about the info you have to display or make available - company number, registered address, etc.