There is no Unitarian Universalist baptism, per se, and yet it is a practice that was common in our history. Although I attach no salvific quality to this sacrament, or any other for that matter, I do wonder why more is not readily known about it in the UU tradition. Perhaps in some areas this ritual could be recovered from our history and usefully practiced, as I'll explain here.
A few of the established Congregational churches evolved into Universalist societies, but the greater number of Universalist churches were made up of the lower classes of New England society. Universalists were come-outers from many denominations; an amazingly large number came from the Baptists. They were of little education compared to the Unitarians and could not boast of a well-educated clergy. Indeed, like the Baptists, many deliberately boasted of the uneducated condition of their clergy. In their view, the Holy Spirit operated freely among men and needed not the trappings of the schools. — Universalism in America: A Documentary History of a Liberal Faith
Frustratingly, I've found scant information on those former Baptists in Universalist history. In particular, I would like to know how long immersion baptism continued among them, and if at some point it either became infant baptism, or simply disappeared. Baptists historically have emphasized baptism as an act of obedience to Jesus, and a necessity to join the church, but not required for salvation. Perhaps it faded away once hell was no longer a major consideration. This still strikes me as strange, because for most of its history in the United States, Universalism was distinctly Christian and Bible-centric. Baptism is prominent throughout the New Testament, from the baptism of Jesus to that of the first believers, and in Pauline theology. Consider Romans 6:
1 What then are we to say? Should we continue in sin in order that grace may abound?
The argument Paul had been building up to this point is that the grace of God covers sin. The rhetorical questions asked here assumes that someone would suggest more sinning so that more grace could fill the world.2 By no means! How can we who died to sin go on living in it?Drawing a clear line here, Paul speaks of turning to Jesus in faith as a death. As the dead do not carry on doing what they did in life, so the believer is no longer to live as they did before coming to faith.3 Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death?Contrary to the majority view among evangelicals in our time, the line that Paul draws is not a prayer to 'receive' Jesus. Instead, it is baptism that ends the former life of the believer.4 Therefore we have been buried with him by baptism into death, so that, just as Christ was raised from the dead by the glory of the Father, so we too might walk in newness of life.The imagery here should be clear. The person lowered into the waters is 'buried' therein, and so when they are brought back up they are rising to their new life. This is consistent with the meaning of the root word for baptism in Greek, which means only immersion, and not sprinkling, pouring, or any other conducting of water. It is like the Jewish mikveh, something already practiced in the time of Jesus, as attested through archeological discoveries. The Orthodox churches continue to practice immersion, and this includes the baptism of infants.
What might be drawn from this is simply that baptism can be conceived of as a death to the old life, and rebirth to a new one. While Unitarian Universalism is itself agnostic, congregations and individuals hold to a variety of beliefs. Given the presences of Baptists in the roots of Universalism in the United States, it shouldn't be considered a stretch for those who see the symbolism as meaningful to want to take it for themselves. As I see it, a UU who had thought this out and wanted to be baptized in this way for personal reasons would not be out of line to seek UU clergy to help them with it. There is, of course, no guarantee that a UU minister amenable to the practice would be readily available. My suggestion is that clergy not reject it out of hand.
The meaning invested into it among us would not have to follow the traditional formula of 'Father, Son, and Holy Spirit,' although it could. Among progressive Christians there are some who baptize 'in the name of the Creator, Redeemer, and Sustainer,' or some variation thereof. While this seems inappropriate for orthodox Trinitarians, as it expresses Modalism, there would be no reason for a theistic Unitarian Universalist to embrace it. Another option is to baptize simply 'in the name of Jesus,' as the Book of Acts tells us was done in the earliest church (what that means is a point of argument, but it's irrelevant for this immediate topic). Finally, any other formulation or none at all could be used. The clergy person could work with the person to be immersed on what would be the most helpful to hear.
So far I've discussed this as an element in early American Universalism, a practice of the early church, and as a personal matter. One other possibility, given the free nature of our UU societies, is that a local congregation could simply have it as a practice. Suppose the unimaginable happens and a Baptist church were to join us (no one would have predicted what happened in Tulsa, so don't giggle too much). We would have no ground to argue against their practice of baptism, so long as they upheld the principles and general theology of Unitarian Universalism. Our expectation would be that no creed would be required for members, and so it's possible that even here baptism could die out. Perhaps that's what happened in the Universalist tradition already.
There is room for greater diversity of belief and practice within UUism, and richer resources than we usually allow ourselves to draw from.