Companies House only hold records of limited companies. If your barber is operating as a sole trader, then his business won't be listed.
HMRC simply require that he keeps a record of his income and his outgoings. He doesn't need to have any form of cash register, he can just keep his money in a tin if he so chooses. As long as he records how much is in there, as a float, at the start of each day (or even at the start of each week) and how much is there at the end of each day (or week), his record-keeping will be sufficient to pass any basic tax checks. There's no need for him to show that he got £17 from cutting Fred Bloggs' hair, £14 from cutting the hair of Fred's young son and £25 from giving John Smith a hair cut and a wet shave. As long as his total income looks to be reasonable for someone in his line of business, HMRC will probably accept it as being correct (irrespective of whether it actually is or not).