How is the soul of a non-Hindu processed in the afterlife? Is it as decribed in Hindu scripture, or is it as described in the scriptures of the religion that they belonged to?
Hardly that they are processed as described in the scriptures of the religion that they belong to.
Teachings of any tradition, be it within Hinduism or outside of it, is not true if it is not in conformity with Hindu dharma teachings.
This is stated in the Manu-smriti, chapter 12:
The Veda is the eternal eye of the manes, gods, and men; the Veda-ordinance (is) both beyond the sphere of (human) power, and beyond the sphere of (human) comprehension; that is a certain fact.
All those traditions (smriti) and those despicable systems of philosophy, which are not based on the Veda, produce no reward after death; for they are declared to be founded on Darkness.
All those (doctrines), differing from the (Veda), which spring up and (soon) perish, are worthless and false, because they are of modern date.