No. It is not a hymn sung at Mass. I know the difference between the Te Deum and the Benedictus. How can you say it is not said/sung at Morning Prayer and then go on to say it is said/ sung at the Office of Readings? We sing it at Westminster Cathedral at Morning Prayer except in Advent or Lent, but never at mass. It is very occasionally sung solemnly on a very special day, but separately after mass.
*GENERAL INSTRUCTION OF THE LITURGY OF THE HOURS:
Chapter II-III. Office of Readings
68. On Sundays outside Lent, on days within the octaves of Easter and Christmas, and on solemnities and feasts the Te Deum is said after the second reading with its responsory but is omitted on memorials and weekdays. The last part of this hymn, that is, from the verse, Save your people, Lord to the end, may be omitted.
Chapter II-IV. Vigils
73. Further, since in the Roman Rite the office of readings is always of a uniform brevity, especially for the sake of those engaged in apostolic work, those who desire, in accordance with tradition, to extend the celebration of the vigils of Sundays, solemnities, and feasts should do so as follows.
First, the office of readings is to be celebrated as in The Liturgy of the Hours up to the end of the readings. After the two readings and before the Te Deum canticles should be added from the special appendix of The Liturgy of the Hours. Then the gospel should be read; a homily on the gospel may be added. After this the Te Deum is sung and the prayer said.
- ARRANGEMENT OF THE OFFICE FOR SOLEMNITIES
- In the office of readings, everything is proper: the hymn, the antiphons and psalms, the readings and the responsories. The first reading is from Scripture; the second is about the saint. In the case of a saint with a purely local cult and without special texts even in the local proper, everything is taken from the common.
At the end of the office of readings the Te Deum and the proper prayer are said.
- ARRANGEMENT OF THE OFFICE FOR MEMORIALS
235 d. in the office of readings, the Scripture reading with its responsory is from the current cycle. The second reading is about the saint, with a proper responsory or one taken from the common; if there is no proper reading, the patristic reading for the day is used. The Te Deum is not said.
The instructions say that the Te Deum is said during the Office of Readings as prescribed above. It clearly states that the Te Deum is said during the Office of Readings; it makes no mention of it being said during Morning Prayer.
You are correct in saying that the Te Deum is not said during Mass, but in my parish it is usually sung on great solemnities immediately following Mass. It’s also usually sung at the end of our two Eucharistic processions: one at the end of our 40 Hour Devotion procession and the other at the end of our Corpus Christi procession. We also sing it at the end of our non-Eucharist procession on our Patron Saint’s Feast day. Like I said earlier, I haven’t heard it sung in Church after mass, since our Eucharistic Procession, in November, at the end of our 40 Hour Devotion.
EDIT: The only time I can think of when I heard Te Deum sung at morning prayer, was when the Office of Readings was combined with Morning Prayer, meaning one was said after the other.
What can I say, we’re pretty traditional in our liturgies, and our “children’s choir” (everyone between the ages of 10 and 23…except me, of course) enjoys singing the traditional hymns and in Latin.