When was plastic invented?

I haven't seen the movie recently so I don't remember the exact line. It depends if she was saying plastic the noun or plastic the adjective. As an adjective, "plastic" has been used for a long while to mean flexible, moldable, or changeable. That's why plastics are called plastics, because they are made of a substance which is plastic (the adjective), or has plasticity.
Celluloid, a very early plastic (the noun), was known to science by the 1860s, and someone which a scientific background might have used the term as a noun. As far as I know, however, it wasn't in common use until after WWII.

She used "plastic" (adj) to describe babies' faces.

Google tells me:
Origin of PLASTIC

Latin plasticus of molding, from Greek plastikos, from plassein to mold, form
First Known Use: 1632

I thought she called the baby's expressions placid

In this context, "plastic" means "moldable" or "capable of assuming different forms." The substances we call plastics are derived from that definition.