There is no specific law saying that a business can't use a name which is already in use. For example, there are taxi (and private hire) firms all over the country trading as 'A2B' and I'd be prepared to rake a bet that there are loads of duplicated names for hairdressing salons as well. (e.g. 'A Cut Above' is probably used in lots of towns). Driving schools provide a further example, with the same names being used in lots of different towns.
However you need to consider the geographic area where you will be trading. If there's another business with the same (or even just similar) name as yours operating in the same area, you could fall foul of the law on 'passing off'. For example, if there's a well-established hairdressing salon called 'Curl Up' operating on one side of a town, someone opening a salon called 'Curl Up & Dye' on the other side of the town might be accused of trying to take customers from the existing firm through pretending to be a new branch of it