You will find, if you try to go abroad, that the authorities will be suspicious of a situation where a child is travelling with only one parent. You would probably be asked to show a letter from the other parent giving consent. A child's name may be looked up on a list of wards of court. If the child's name is found there, the father may well be arrested.
You can, however, go to court and get the child named as ward of court, and get the court's permission to take him on holiday. This will outweigh the mother's objections, but it can take ages, cause endless stress, and cost a fortune if you use solicitors. (You don't have to use solicitors, by the way.)
Perhaps the mother should just be reminded that if she does refuse the child a wonderful opportunity like this, he may not forgive her.