You could google your way to a lot of facts about the very many different days on which this Feast has been kept, as 'Gallican' bishops, and, later, Popes, granted it to various places or orders. The earliest date appears to be October 20; a date which enjoyed the favour of S John Eudes as early as 1647. But there seems to be an increasing tendency to fix it on the Octave Day of the Assumption; or the Sunday within ... or the Sunday after ... that Octave. My 1874 Breviary has it then (among, of course, the observances For Various Places).
But liturgical books which I have, dated 1922 and 1957, assign it to the Saturday after the Feast of the Sacred Heart. My instinct is that this might be a result of the reforms of S Pius X or of the Message of Fatima. Perhaps readers may be able to pin this down.
In 1942, "gravissimas miseratus aerumnas quibus christiani populi ob ingruens immane bellum affliguntur" Pius XII consecrated the Human Race to the Immaculate Heart, and ordered its feast to be kept in the Universal Church on August 22 instead of the old Octave Mass of the Assumption. It will be seen that this represented, in calendar terms, a reversion to the nineteenth century date of the Feast of the Most Pure Heart. The Bugnini idea of using the Saturday after the Feast of the Sacred Heart was, you will have noticed, not a totally new idea. It was a reversion to the date employed in the first half of the twentieth century for the feast when our Lady's Heart was described as 'Most Pure' rather than as 'Immaculate' and was still pro aliquibus locis rather than universal.
If calendars are one day to be harmonised, it would seem to me appropriate to keep this lovely and Biblical feast on the date after the Sacred Heart, where it was until 1944 and then after 1970, not least because that would fit the 'First Saturday' devotion. The Octave Day of the Assumption, and Maria Regina, could then fight it out for August 22. My instinct would be to call the day by its proper name, the Octave Day of the Assumption, but to incorporate some features of Pius XII's Office or the Liturgia horarum, such as the hymn O quam glorifica, which dates from the ninth century and was originally proper to Assumption Day itself.
May 31 should revert to being the Feast of our Lady Mediatrix of All Graces, as it increasingly was in many places where granted by indult, until Pius XII parked Maria Regina on that date. (When Bugnini wickedly transferred the Visitation to May 31, Dom Lentini did make a feeble attempt to keep a memory of this earlier celebration of our lady of Grace by including, in a hymn he composed for the Visitation, the stanza Teque felicem populi per orbem/ semper, O Mater, recitant ovantes/ atque te credunt Domini favorum/ esse ministram.)