Is it reasonable for a copy shop to refuse to make a copy of a photograph of my mother taken by a professional photographer in 1962?

Probably not.  All photographs taken in 1945 or more recently are still in copyright and are likely to be in copyright at least until the end of 2015, so a copy shop would want to be sure that the copyright owner had agreed to the copying of a photograph. However, if the photograph was commissioned by your mother and taken before 1989, she would have been the first owner of copyright unless there was an agreement to the contrary. Your mother's heir(s) probably own copyright now and they could authorise the copying. (You should note that people who have commissioned photographs since 1 August 1989 are not the first owners of copyright.)