As far as I am aware there isn't usually a way of adding it to the birth certificate after 12 months.
However, I think if you book an appointment at local office and take evidence that you have been using a middle name as well then that may be the exception. Evidence can be anything, or so I was told- letters etc.
In the event that they do not agree to that you can add a middle name by deed poll, which he keeps with his birth certificate- he'll be used it when he's older and will probably prefer that to not having a middle name. It's very easy- uk deed poll service is good and fast.