We learn in the very first act that Juliet's nurse became her wet nurse at the time of Juliet's infancy after having lost her own daughter, Susan, who had been the same age as Juliet, as we see in Nurse's lines:
Susan and she (God rest all Christian souls!)
Were of an age. Well, Susan is with God;
She was too good for me. (I.iii.22-24)
Hence, since Nurse's own daughter had died soon after her birth, Nurse was still lactating for her own daughter and was able to feed Juliet.
A wet nurse is a lactating woman who is hired to nurse the baby in lieu of the mother nursing the child. Wet nurses were common practice in the time period in which Romeo and Juliet is set in, especially for members of the Catholic Church. Lactation is a form of birth control, and the Catholic Church deemed it sinful to have intercourse without the intention of procreation. The Church therefore banned intercourse while nursing. Wealthy women avoided being accused of breaking the ban by simply hiring wet nurses (Day, "Wet-Nursing and the Catholic Church").
While it may not have been common practice to continue keeping a nurse on staff to be a daughter's companion while she grows up, it certainly was common practice for wealthy girls to be raised and educated by a governess. It may be that Nurse was kept in the household as Julet's governess. At any rate, we know that Nurse has stayed on as Juliet's companion rather than still as her wet nurse. We especially know she is no longer Juliet's wet nurse because Nurse tells a story about weaning Juliet in the third scene of Act 1.