Will I be able to get a refund when …

Let’s say I purchase two non-refundable one-way tickets on different airlines instead of one round-trip ticket. My outbound flight is cancelled due to weather and I chose to take a refund. Will I be able to get a refund on the return flight which is on a different ticket and airline?

No. Separate tickets are treated as individual transactions unrelated to one another and so the second airline would have no obligation to create and exception and refund their ticket. You could certainly cancel that flight before the time it was due to depart and retain the value of that ticket, minus the airline change fee, for future use.