To receive child benefit in the Irish state,
a) The child must be ordinarily resident in the Irish State
b) The applicant (usually mother) must be habitually resident in the Irish State (which can be hard to establish if you retain financial links to other countries, even to the North)
However, if the applicant/mother is an EU national (ie, other than Irish) then she will be deemed to satisfy the habitual residence condition.
In other words, you should be eligible for child benefit from the Irish state. The usual rule is that you apply for the benefit in the country where you work. If your wife is working in the UK, she should apply to the UK authorities to let them know that the child is now in Ireland, and they will contact the Irish authorities to ensure that you get the difference topped up. If your wife is not working, then she should apply directly to the Irish authorities here, and obviously let the UK authorities know that both she and the child are no longer living in the UK.
If you or your wife are working in UK and are living in Ireland you will also need to inform the Irish Revenue as you are likely to be considered resident in Ireland for tax purposes and will have a tax liability here, although they will deduct any tax that you pay to the UK authorities from your tax liability.