You've been given the correct advice above, but here is a little more detail. Presuming you live in England, you need to apply to the court for a Specific Issue Order to change the name. Change of Name Deeds will be worthless once he is older and needs a driving licence/passport etc, so a Court Order is the only route to do this formally. His dad will need to be served with the application (this means he needs to be sent a copy and told about the time, date and location of the hearing). You can tell the court that you only have his mobile number but they are likely to require you to find his address to serve him properly. This can be done by applying to the same Court for an Order from the Department of Work and Pensions to disclose his address for the purpose of serving him. Once served, he will be entitled to play a role in the proceedings if he wishes, but he doesn't have to. keep in mind, he could also make his own application to the court, for example for contact with your son, if he wanted. I don't think you say how old your son is. This is a relevant factor to any decision that the court will make.
The Court's overriding deciding factor, will be your son's welfare, and the impact of the change of name on that. Google the "welfare checklist" if you want to know what factors they will consider.
This is a really brief outline of the procedure/law and it would be a good idea for you to have half an hour of free legal advice before you go down this route, if you decide to do so. Going to court can sometimes risk 'opening a can of worms'. Also, I believe the Court system currently has a backlog of many months for private family applications (although this will vary from Court to Court) so don't expect this to be quick. You are probably looking at a good few months, especially if there is a delay in getting an address to serve him. If you can find this yourself, you will save a lot of time. Good luck!