Welcome Shadrack.
The nut falling off is not uncommon nor a big deal especially with a Gibson-style nut/nut slot. I would think there have been more than a few temperature and humidity changes in its journey from Australia to BC!

Barry snook makes a good point about upgrading the stock plastic nut to a better quality. Replacement nuts are not expensive and there are a range of materials to choose from. Bone is very common, but there is GraphTech or synthetic bone materials as well. It's all personal preference. If you decide to upgrade, we can offer more specific advice on the process when you're ready.

As for cleaning up the old glue in the nut slot, I recommend using single edge safety razor as a scraper. Hold the blade perpendicular to the area you're working (so you're scraping not slicing IYKWIM).
I recommend this instead of a file or sandpaper because you have more control to remove the just the glue and not wood in the slot.
You don't want to inadvertently widen or deepen the slot or make the bottom uneven.

When you glue in the old or new nut, use a wood glue like Titebond I (sometimes called "carpenter's glue" - do not use super glue!) and only a few strategically placed drops applied with a toothpick. Make sure the nut is seated and centred properly and then lightly clamp it until the glue is set.
You literally only need enough to hold the nut in place when the strings are off for changing or maintenance, and if it needs removal again, it can be knocked out with some gentle taps and not a BFH (Big Freakin' Hammer ).


I will catch hell from some for saying this, but for nuts, I am fine with using a PVA glue. Other jobs, Titebond (Aliphatic Resin Emulsion) is more appropriate.