The patrons of children:Bathilde
Blaise
Gabriel
Gowdel
Gerard Majella
Gezelin of Schlebusch
Holy Innocents
Infant Jesus of Prague
Joseph
Maria Goretti
Nicholas of Myra
Nino de Atocha
Pancras of Rome
Philomena
Raymond Nonnatus
Solange of Bourges
Symphorian of Autun
Trophimus of Arles
Ultan of Ardbraccan
Vaast of Arras
As per the Patron Saint Index, William of York is a saint but not currently a patron saint.
There is no patron saint of bullying or against bullying.
Yes he is the patron saint of repented thieves.
There is no patron saint of children's health but the Infant of Prague is the patron saint of good health.
The patron saints of unborn children are Saint Joseph and Saint Gerard Majella.
Saint Nicholas of Myra is the patron saint of children, scholars and penitent thieves as well as a number of other causes. For a complete list of his patronages, click on this link.
As per the Patron Saints Index, the patron saints of unborn children are Saint Gerard Majella and Saint Joseph.
I see no specific saint for healing of children. However, St. Brigid of Ireland is considered as the patron saint of healers.
The patron saint of American soldiers is the Immaculate Conception, The Blessed Virgin Mary
The patron saint against autism is St. Ubaldus Baldassini.
There are patron saints of children or orphans or social workers but no patron saint of child custody.
The patron saint of running water is St. John Nepomucene