Current law requires requires that your dog wear a tag with your name, address (including postcode) and ideally phone number.
"The Control of Dogs Order 1992 mandates that any dog in a public place must wear a collar with the name and address (including postcode) of the owner engraved or written on it, or engraved on a tag. Your telephone number is optional (but advisable)." [ref Kennel Club website]
If you fail to comply with the law then you risk a fine of up to £5,000. With the wide variety of tags available there's no reason for even the most fashion conscious pup not to wear a tag.