How old was Moses when he went to live with Pharaoh's daughter?
Most likely just after being weaned from the breast.
Pharaoh's daughter said to her, “Take this baby and nurse him for me, and I will pay you.” So the woman took the baby and nursed him. When the child grew older, she took him to Pharaoh's daughter and he became her son. She named him Moses, saying, “I drew him out of the water.” Exodus 2:8-10 NIV
Since the princess was only paying for nursing services and she wanted a son badly, once her milk dried up baby Moses goes back to the new Pharaoh's daughter mommy at the ripe old age of 6 months-ish.
The answer of 6 months is a good start as scientists say that this is the time the immunity has built up enough to have benefited the child to guard against diarrhea and respiratory tract infection. However, I doubt this has been the practice historically and in a culture which prizes women for their roles as nurturers. Judaism has a usual age recommendation of 24 months but as long as 4 years of age, which makes sense as a child of that age would be on solid foods. There is a prohibition against being off breast milk for 72 hours and then resuming but the child can still drink milk from a "bottle" at that point.
Egyptians had a culture that had some similarities to the Hebrews and other cultures of the region. Here is a source backing what I wrote about weaning in ancient Egypt
During the first years of life children are busy acquiring basic skills like walking and speaking and play no economic role. Their chances of survival improved when they were weaned late , as in a hot country like Egypt diseases of the digestive tract are widespread.
 The age of three years seems to have been a common age for weaning. At times older children were referred to as still suckling, apparently hyperbolically. The mortuary stela of Isenkhebe speaks of death as The dark, a child's terror, engulfed me, While the breast was in my mouth! From the Stela of Isenkhebe, 7th century BCE M. Lichtheim Ancient Egyptian Literature, Volume III, p.59
Data collected from modern non-industrial societies suggest that infant mortality in ancient Egypt was undoubtedly high. One of the best ways to maintain a healthy infant under the less-than-sanitary conditions that prevailed in ancient times was by breast-feeding. In addition to the transfer of antibodies through mother's milk, breast-feeding also offered protection from food-born diseases. Gastrointestinal disorders are common under poor sanitary conditions, and because infant immunity is reduced during weaning, children's susceptibility to disease increases at this time. Indirect evidence for this occurring in ancient Egypt comes from a number of cemeteries where the childhood death rate peaks at about age four, which correlates with an Egyptian child's introduction to solid foods. Prolonged lactation also offered a number of heath advantages to the mother. Primarily, it reduces the chance of conceiving another child too soon by hormonally suppressing ovulation, which allows the mother more time between pregnancies. The three-year period for suckling a child recommended in the "Instructions of Any"