Hi Chemmie,
We would like to convey the meaning that the baby will be away from his mom. The preposition that we should be using is "from", which is used to convey a meaning of separation through space or time.
For example:
Los Angeles is a five-hour drive FROM San Francisco. (Separation through space)
Please isolate the patient FROM the other patients in the ward. (Separation through space)
This operation will take place from 9 a.m to 3 p.m. (Separation through time)
Therefore, your third sentence is correct. Since the baby is being separated from his mom, we should write: It's the first time that the baby is separated from his mom.
*If you want to use "with", we can rewrite sentence 1 as: "The baby is made to part with his mom for the first time."
In this case, "with" conveys the meaning of relationship or an association.
Example: I love tea with cookies.
Hope this helps.
Cheers, Lance