What you've got there looks a lot like the old
flax wheels that turn up around here (northern Sweden) quite often.
I've picked up a few and worked on making them useable, with varying levels of modernisation. If we lived nearby, I'd be delighted to help get you started. As it is, I can offer encouragement and advice as you go along.
I've added pictures of two of my recent projects: they started as a pair of identical (almost destroyed) wheels. One has been returned to almost original style, with a larger orifice grafted onto the original flyer.
The other is now my favourite; I wanted a large capacity and more compact style, so I used the parts to build a Louette-type wheel and made a complete bobbin and flyer from scratch.
On both wheels, all the original parts are blue and the rest I've made myself.
I could ramble in, giving all the tips I learned from these adventures, but don't want to hog the space and of course you're very welcome to send me a moosage any time 🤠
I haven't had an axle come loose like that, but all the wheels I've seen here have been dry-fitted, or with the occasional dab of hide glue, so it's not unexpected. If you have engineering skills, apply them with care and you'll be fine.