Easter Vigil is the strong preference. Sunday, certainly, for the conferral of the initiation sacraments. But Roman pragmatism allows for exceptions:
306. If the sacraments of initiation are celebrated at a time other than the Easter Vigil or Easter Sunday, the Mass of the day or one of the ritual Masses in the Roman Missal, “Christian Initiation: Baptism” is used. The readings are chosen from those given in the Lectionary for Mass, “Celebration of the Sacraments of Initiation apart from the Easter Vigil”; but the readings for the Sunday or feast on which the celebration takes place may be used instead.
It seems clear that the connection with the “remembrance of Christ resurrection” (RCIA 304) would move this “votive Mass” to a higher plane where most all Sundays are concerned. The choice to use special readings or the readings of the day would be determined by the requirements of ministry, not the personal choice of the pastor or liturgical ministers.
Those baptized companions are considered seriously important, as we see here:
307. All the children to be baptized are to be accompanied by their own godparent of godparents, chosen by themselves and approved by the priest (see no 11, Christian Initiation, General Introduction, no. 10).
308. Baptized children of the catechetical group may be completing their Christian initiation in the sacraments of confirmation and the eucharist at this same celebration. When the bishop himself will not be the celebrant, he should grant the faculty to confirm such children to the priest who will be the celebrant (Rite of Confirmation, Introduction, no. 7b). For their confirmation, previously baptized children of the catechetical group are to have their own sponsors. If possible, these should be the persons who were godparents for their baptism, but other qualified persons may be chosen. (Rite of Confirmation, Introduction, no. 5-6)
I don’t know about your parishes, but I’ve never served in a community where celebrations of Confirmation and First Eucharist were included in the RCIA initiation rites. More usual is for children to be baptized with adults. I can imagine there would be a sense of disconnect if non-Catholic children were confirmed and First Communion candidates of the same age were not. And yet the suggestion in the rite is undeniable.