There will be no formal record. If he moved to the opposite end of the country and appeared there as Harry Lancaster no-one would have known he had been baptised as Harry Johnson. Yes, children may have taken their father's name if their parents subsequently married, the same as children often appear in censuses under their step father's name, and, less often, married under the step father's name.
Proving it was the same person is another matter altogether. It's hard enough to show that someone in the next village with the same name is the same person, let alone someone who appeared in Scotland under a different name. Unless you can find him in a census with the same birthplace, and even then proving it is the same person is a challenge.
David
