Suppose that option is not available, say the nearest maternity hospital is 100 miles away.
What do you do next, sit back and let the Imam perform the ceremony?
I’m hard pressed to think of one Muslim hospital in the USA, much less one that’s 100 miles from the nearest ‘maternity hospital.’ And I’m hard pressed to think of anyone who’s expecting a baby who hasn’t made plans about the delivery hospital long in advance of the birth.
I would think it highly unlikely in this day and age for staff at a Catholic hospital to baptize a child, or that a dying child would be in the hospital without his/her parents at his/her bedside, except in the case of a severe emergency. In a case like that, an unidentified dying child might be baptized in a Catholic hospital’s ER. But I don’t think Jewish parents of a sick child who take their child to a Catholic hospital have anything to worry about.
They might be more worried that their child, if they are Orthodox, not get very sick on the Sabbath. My sister lived in an apartment complex with a very large percentage of strictly observant Orthodox Jews. One family let a child die because they would not pick up the phone on the Sabbath.