If you look in the good old English Missal, you can find our Lady as Mediatrix of all Graces in the Appendix of Masses Proper to England and Wales. This is because in Durham, Northumberland, the West Riding of Yorkshire, Herefordshire, and all Wales, this feast was on the Calendar of the corresponding RC dioceses. The date is May 31, except that in later editions of the English Missal, dating from after the institution in 1955 by Pius XII of the feast of the BVM, Queen, on May 31, the older feast of our Lady, Mediatrix, had to be shuffled onto June 1. The Mass can, of course, be said as a votive any day when votives are permitted.
Happily, one of the Office Hymns of this beautiful Feast appears in the Liturgia Horarum. In the Common of the BVM, it is the hymn at First Evensong; and it may be used at the Office of Readings as an alternative to Quem terra pontus aethera. Its first line is "Maria quae mortalium".
Sadly, for those who say the Office in English, it is not available; it is one of the many hymns for which ICEL decided not to bother to commission English translations. (Whereabouts did Sacrosanctum Concilium sanction this wholesale disparagement of the treasures of Christian Latin Hymnography?) It may be found, with an English version, in versions of the Monastic Diurnal for the Feast of our Blessed Mother of Perpetual Succour. Its (nineteenth century) author is not known.
The Mass and Office were authorised in 1921 by Pope Benedict XV, and among the countries for which it was authorised was, I believe, Belgium. It had been requested by that dear old friend and doughty supporter of the Anglican Patrimony, Cardinal Mercier, Archbishop of Malines. It is my belief, which I am unable to support with evidence, that Papa Pacelli in 1955 deliberately put Maria Regina onto May 31 because he was suspicious of the doctrine of her Universal Mediation and wished to repress the growing enthusiasm among dioceses and countries to petition for this feast.
So the whole Latin Church (EF) now has Maria Regina today. I feel the Pius XII propers are a trifle unsophisticated; they emphasise the old literary topoi of Royalty. No harm in the that. But the older liturgical understanding of Mary's glorification, in both East and West, connected it with her role as Mediatrix for all the Church. The only element in the Pius XII office which relates to this is lectio ix, where S Bonaventure proclaims Mary as the New Esther and as the aquaeductus through which the Grace of God comes to us.