There is a difference between the UK and Ireland (as opposed to the UK and Oz) as they are both in the UK-Ireland Common Travel Area and both in the EU (obv that will change). UK-Australia child custody and visitation and residence issues are governed by the Hague Convention.
europa.eu/youreurope/citizens/family/children/parental-responsibility/index_en.htm
Going to court over custody and visiting rights:
If you cannot reach agreement on custody or visiting rights, you will probably go to a court.
In situations involving more than one country, for example if the parents do not live in the same country, the courts responsible for handling cases of parental responsibility are those in the country where the child usually lives.
If you expressly agree with your spouse, the court ruling on your divorce can also rule on parental responsibility matters connected to your divorce.
The court will decide in the best interests of the child on your custody rights, visiting arrangements and determine the child's place of residence.
Recognition and enforcement:
Court decisions on parental responsibility made in an EU country are recognised in all EU countries without the necessity for any additional procedure.
Their enforcement is facilitated by a standard procedure.
eur-lex.europa.eu/legal-content/EN/TXT/?uri=LEGISSUM:l33194
The EU legal context explained here.
Matrimonial and parental judgments: jurisdiction, recognition and enforcement (Brussels IIa)
As a general rule, this regulation replaces existing conventions on the same matters involving 2 or more EU countries. In relations between EU countries, it takes precedence over certain multilateral conventions:
1961 Hague Convention (law applicable to protection of minors)
1967 Luxembourg Convention (recognition of decisions on marriage);
1970 Hague Convention (recognition of divorces)
1980 European Convention (custody of children)
1980 Hague Convention (civil aspects of international child abduction).
With regard to the Hague Convention of 19 October 1996 on parental responsibility and measures for the protection of children, the regulation is fully applicable if the child normally lives in an EU country.
@Mommy273, you absolutely must see a solicitor and get an arrangement in place, preferably before Brexit happens (I believe the date is March 29th 2019).
Pick up the phone ASAP and get this done.