"We save the mother on time" suggests a certain deviousness. It suggests planning. "On time" could mean "on schedule, as planned, as intended". That works with constructions like "The bus/train left on time" or "The play started (right) on time".
"In time" implies "before the opportunity to do so was wasted/unavailing/useless".
"We arrived in time" suggests "before it was too late". "We arrived on time" suggests "at the right time to participate".
But that would need different cases: "We save time for the mother " (i.e., we save time for her benefit): time would be accusative, WHAT we save ( = Latin tempus); mother would be dative, TO/FOR WHOM we do the action (Latin matri).
"Time" is a 'weird' word, being a neuter of the 3rd decl: tempus, temporis, neuter.