I don't know about a database, but do you have the book edited by John Lind (1965) Newborn Infant Cry? However, this is different from what you are looking for--they tried to record the babies in the first few hours of life, not distinguish among cries. They also tied the cry sounds to cinefluorography of the oral structures.
couldn't find any standard database but yes i guess this would be helpful: https://opendata.stackexchange.com/questions/6771/required-a-audio-format-baby-crying-data-set