Sure.
The Solemnity of the Assumption does take precedence over the 20th Sunday of Ordinary Time. So the liturgical day of the Assumption begins (began) on Friday evening and ran until midnight Saturday night.
Strictly following the calendar, all Masses on Saturday evening should have been Masses for the Solemnity of the Assumption.
However, in 1974 the Holy See addressed the issue that there can be confusion (exactly the sort that’s being expressed in this thread). The response was that there should be a pastoral approach. The conclusion was that even though the calendar requires that the Solemnity replace the Sunday, a certain flexibility is allowed. See the last sentence of the text I’m about to quote.
Here’s the full text:
source:
notitiae.ipsissima-verba.org/
The Instruction Eucharisticum mysterium, n. 28 established: « Where permission has been granted by the Apostolic See to fulfill the Sunday obligation on the evening of the preceding Saturday … the Mass should be celebrated as assigned in the calendar for that Sunday, including the homily and the prayer of the faithful.
The same must be said of a Mass which, for the same reason, is permitted to be celebrated on the evening of the day before a holy day of obligation in a particular place ».
This is the general norm. A doubt, however, has arisen when a certain obligatory solemnity occurs on a Saturday or a Monday. For on the evening of the first feast day (Saturday or Sunday) there is an overlap of liturgical days because « the celebration of Sunday and of solemnities begins already in the evening of the proceeding day » (cf. Normae universales de anno liturgico et de calendario, n. 3), and in the same celebration some of the faithful come to satisfy the precept pertaining to the current day and others that which pertains to the following day. Thus, for example, it can happen that the on the evening of the Fourth Sunday of Advent, when this falls on December 24, there would be at the same time the evening Mass of the Sunday and of the Vigil of the Nativity of the Lord. In the same way, when the Nativity of the Lord falls on Saturday, there is on that evening there occurs an overlap between the Mass of the Nativity and the anticipated Mass of the Holy Family.
This and similar cases cannot be solved by the universal norms, because there exist different pastoral necessities and customs of the faithful.
For that reason the following indications can be given:
- The general principle for the celebration of the Mass of a holy day of obligation anticipated on the evening of the preceding day is that which is found in n. 28 of the instruction Eucharisticum mysterium.
- In the case of an overlap between a Sunday and a Solemnity, the integrity of the celebration of the whole liturgical day can generally be better preserved by applying to the celebration of the evening Mass what is established for Vespers, namely: « If on the same day the Vespers of the current Office and Vespers I of the following day are to be celebrated, the Vespers of the celebration placed higher on the table of liturgical days takes precedence; and in case of equal rank, the vespers of the current day takes precedence » (cf. Normae universales de anno liturgico et de calendario, n. 61).
- On the vigil of solemnities for which a particular vigil Mass exists (Nativity of the Lord, Nativity of St. John the Baptist, Sts. Peter and Paul, the Assumption of the Blessed Virgin Mary), this should be said even if it falls on a Sunday.
- Attentive circumstances of a pastoral nature, the practice to follow at the diocesan level should be indicated by the local Ordinary at the beginning of the year in the proper liturgical Calendar, even, if the case warrants, by derogating from what was said above when, for pastoral reasons, the celebration of one or the other Mass seems preferable.
The underlining in that last sentence is my own.
The practical application is that even though it might derogate from the rules of the calendar, it is permitted to celebrate the Mass of 20th Sunday on Saturday evening even though the norms say that it should be the Mass of the Assumption.
Bottom line: the priest should follow the liturgical calendar published by the diocese.