There are two separate requirements:
1. Nominet gives bona fide non-trading individuals the right to have their address (but not their name) excluded from the database of owners of .uk domain names. (They have to request such exclusion). Everyone else has to accept that, by registering a .uk domain name, their details will appear on a publicly-accessible database.
However, while Nominet is an 'official' body (in that it's the one authorised to allocate .uk domain names), its rulings aren't part of UK law.
2. The Electronic Commerce (EC Directive) Regulations are derived from the much (and often falsely) maligned European Commission, with that body's rulings being incorporated into UK law through the Regulations. They don't say anything at all about whether the correct address should appear on the Nominet database but they do demand that the normal trading address of any 'sole trader' business must appear on that business's website. (For a limited company it's their 'registered address', as listed at Companies House, which must appear).
So the trader is breaking the law by not including his business address on his website.