There used to be an official Register of Business names,started in 1916, the purpose of which was to identify proprietors of businesses and to minimise the confusing and accidental duplication of trading names, but it was abolished by one of the Companies Acts. There were eventually just too many businesses with the same or similar names (e.g. Jones the Butcher; Smith and Son ) to make the Register useful.
I fear that intending hairdressers have to indulge their fondness for puns (e.g.Curl Up and Dye) in the simple hope that nobody has the same one in such circumstances that their one will be stopped by a 'passing off' action or claim of infringement of copyright.