Canon 1246 of the Code of Canon Law of the Catholic Church states:
Sunday, on which by apostolic tradition the paschal mystery is celebrated, must be observed in the universal Church as the primordial holy day of obligation. The following days must also be observed: the Nativity of our Lord Jesus Christ, the Epiphany, the Ascension, the Body and Blood of Christ, Holy Mary the Mother of God, her Immaculate Conception, her Assumption, Saint Joseph, Saint Peter and Saint Paul the Apostles, and All Saints.
With the prior approval of the Apostolic See, however, the conference of bishops can suppress some of the holy days of obligation or transfer them to a Sunday.
The dates on which various feasts of the Church are held are primarily contained in the Calendarium Romanum, the Roman Calendar, which was last updated by Pope Paul VI in 1969. I can only find a copy in the original Latin; it reads in the section about the Epiphany and Baptism of the Lord:
In Epiphania Domini, die 6 ianuarii, nihil innovatur. Attamen ubi haec sollemnitas non est de praecepto servanda, transfertur ad dominicam post octavam Nativitatis (id est, a die 2 ad diem 8 ianuarii occurrente).
Dominica post diem 6 ianuarii, festum Baptismatis Domini celebratur. Quando vero hoc festum occurrit una cum sollemnitate Epiphaniae ad hanc dominicam translata, illis in locis omittitur.
Nothing is changed regarding the Epiphany of the Lord, on the 6th day of January. However, where this solemnity will not be observed according to tradition [i.e. on the 6th], it is moved to the Sunday after the Octave of the Nativity (that is, to the Sunday occurring from the 2nd to the 8th day of January.
On the Sunday after the 6th day of January, the feast of the Baptism of the Lord is celebrated. When, however, this feast occurs together with the solemnity of the Epiphany moved to this Sunday, in those places it is left out.
It appears that a later ruling decided to move the Baptism of the Lord to the Monday following the celebration of the Epiphany in such a case, rather than to leave it out; but I can't find documentation of this.
Thus, the current rule is this:
The governing episcopal conference of a territory, with the approval of the Holy See, will decide whether they wish all parishes within that territory to celebrate the Epiphany on the 6th of January (the traditional date), or on the Sunday following the 1st of January. These are the only two possibilities, and all parishes within the conference must adhere to the one chosen by the bishops.
If the conference decides to celebrate the Epiphany on the 6th, then the Baptism of the Lord is celebrated on the following Sunday (the 7th to 13th).
If on the other hand they choose to celebrate it on the Sunday after 1 January, then:
- If in a given year this Sunday falls on or before the 6th, the Baptism of the Lord is celebrated on the following Sunday.
- But if the Sunday on which the Epiphany is celebrated falls on the 7th or 8th, the Baptism of the Lord is celebrated the next day, Monday the 8th or 9th.
It appears this is what is happening in your example. Ireland and Northern Ireland are the territory of the Irish Catholic Bishops' Conference, They appear to have elected to celebrate the Epiphany on the traditional date, 6 January, and the Baptism of the Lord on the following Sunday, 8 January. England and Wales are territorially part of the Catholic Bishops' Conference of England and Wales, which seems to have chosen to celebrate the Epiphany on the Sunday folowing 1 January. This year, that is 8 January, and thus the Baptism of the Lord will be celebrated there on Monday 9 January.
It appears from a brief search of the Anglican Communion website that the Epiphany may be celebrated in Anglican churches on either of the days allowed by the Catholic Church. But it's not clear who has the authority to decide when individual parishes celebrate it, or what the practice is in the Church of England in particular.