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How long does it take for resources to transfer from, for example, a parent planet to a colony? Is it a fixed amount of time or is there a calculation?

I'm still running into issues where colonies aren't receiving goods sent from my homeworld and I can't determine whether I'm just not patient enough, or it's really just busted.

IIRC it's about 20 game days for the Merchant AI to arrive in your system, travel to your homeworld for pickup, move away, FTL to the target system, and travel to the target.

If you use Ctrl+Shift+S to access the console in game, you could turn off Fog of War and watch the Merchant AI travel to track the delivery. Maybe take a save before you adjust that setting, it will reveal a lot of stuff you might not yet have explored.