She gave her pills on schedule.

She gave her HER pills according to a schedule.

or does the Russian imply "on time"?

According to the regimen

I'd say according to prescription.

Uly, 2 times HER in your sentence... does it sound ok?

Ну, как-то странно звучит)

А разве "on schedule" и "on time" - это практически не одно и то же?

When we talk about medication that’s been prescribed, we always use a possessive

And the way we pronounce it, there’s no strange repetition: gaver hr pilz