Ah, weapons, hobby interest !
A crossbow is a reasonably precise bit of machinery, needing reasonable manufacturing precision as well as storing enough energy to take your fingers off it if fails whilst cocked. That may be a plot device you can use ! You'd really not struggle if pulling any aeroplane apart to find control pushrods of one sort or another than could do a reasonable impression of a crossbow bolt. I doubt you'd find them in the seat.
The simplest weapon I know that is easy to manufacture and use is a spear. A spear, contrary to popular opinion, is in most variations not a throwing weapon. Basically it's a stick with a knife on the end of it, and the techniques of using a spear are very similar to those for a longsword. Historically people who weren't important enough to justify the significant expense of a sword, got a spear, which uses the minimum amount of expensive metal. There's plenty of history of effective hunting with spears - and they ARE NOT thrown. Typical length of a spear is 4-6 feet.
Knifes / daggers/ are easily cut and ground from bits of scrap metal - fighting with knives is a messy game, but they're an essential survival tool of course. A knife tied to the end of a stick is a spear! Between two people, attacking somebody with a knife takes little skill or fitness, defending against a knife effectively takes very considerable skill.
For a weapon, the next most straightforward improvised weapon would be a bola. Three weights, three bits of string / wire / rope, tied together.
After that would be a slingshot - easily made from any reasonable strong fabric, ammunition is stones.
I could teach you the basics of fighting with a spear in a few hours. Attacking somebody with a knife in 10 minutes, and defending yourself effectively against a knife in a few weeks.
Using a bola is very easy, and takes little skill, just a bit of thought and practice.
Using a slingshot is a matter of a few hours or days of practice, but is fairly intuitive.
It would take a lot of skill and time to make an effective crossbow.
Incidentally, making a bodgers lathe is pretty straightforward:
http://www.greenwoodworker.co.uk/Archive.htm Where I live, up until maybe 60 years ago, people used to just go out into the woods with an axe, saw and some rope, and come back a at the end of the day loaded up with chairlegs.
G