To begin with, you are very right that upper class families would employ wet nurses, called a nutrix. From epigraphic evidence, we have an understanding that families employed not a single nutrix, but several. This helped ensure that not only the children of the head of household had proper nutrition, but so too the children of servants in the household. There seems to be an opinion among ancient authors that it would be a good thing for mothers to nurse their own children, but this opinion seems to be born of the fact that the opposite was the norm. Children in a wealthy household, regardless of what end of the social spectrum they were born into, appear to have been nursed though a communal effort of nurses.

There is, as far as I'm aware, no evidence of anything like formula in the Roman era. To the best of my knowledge it is a marvel of the modern world. That said, there IS documentation that premastication, chewing food for young infants, was common practice. This would be reserved for infants already weened off breast milk, and again, was likely done by nutrices / nurses.

Breast feeding was certainly a concern for Romans, as is evidenced by the amount of male authors weighing in on the topic (see especially Plutarch). But this went hand in hand with the idea that high infant mortality rates were something to be accepted. In emergency situations, I am sad to say that there were no safety nets for child nutrition.

Edit: sourced from "Wet-nursing in Ancient Rome: a study in social relations" by Kieth R. Bradley, in "The Family in Ancient Rome" Rawson, B. 1992