There are ways to recreate cut threads e.g. using 'followers' but at the end of the day the method for recreating a known but unavailable pitch is called single-point threading and it's an option only really available on metalworking lathes. And not all lathes can cut all threads, since it's a matter of how the various gears work together.
Remember also that the shape of the threading (the thread profile) is also important in a softer material like wood, so this goes beyond the usual concern of 60° or 90°? (90 being preferred for wood for greater durability, even at very very coarse pitches.)
Given the amount of time, effort and experimentation required to recreate each one by other means I think this is a non-starter personally. But I can offer three workarounds.
Use the bottle, or at least a bottle, to cut the threading
Just as you can make rudimentary taps from screws or bolts one can, in theory at least, use the glass to cut its own threads. This is doable these days due to the widespread availability of diamond abrasive tooling for small Dremel-type drills or die grinders. You'd use these to cut multiple flutes in the threading — just one flute one is possibly asking too much of glass — possibly in stages, widening the flutes each time (rather than deepening them).
I think you'd need at least two of each bottle type to make this workable. The bottle to cut the threading being sacrificial since its threads will be permanently altered and this could make it unusable in the normal way.
Form the thread in something other than wood
Drill an oversize hole, smoosh in something that will conform and harden, then screw the cap on and clean up the inevitable squeezeout. Then leave the material to dry/harden. This could make for an easy-to-do and long-lasting solution, adaptable to nearly anything.
The obvious first thought here for me is to use thickened epoxy, but commercial epoxy putties (which I'm often not a fan of for woodworking) could work great for this. If you already have Aurolite or similar adhesive, like epoxy they can be mixed with wood dust to form a paste that is very strong upon curing. Going by its reputation, Durham's Water Putty could be good for this too.
Any hard-setting heat-formable plastic such as Polymorph could be used too I think, although I'm not sure about the bond with the wood.
And last but not least....
You already have something with the threads in it
You already have caps for each bottle, how about just setting them into holes in the wood caps?
A surprising number of commercial wooden bottle caps use this basic method, with an insert going into a much-simpler-to-produce basic wooden turning. Although the threaded portion is usually plastic the basic principle would hold for the existing caps which I'm presuming are mostly thin metal as on screw-on caps for wine bottles.
This is arguably the least elegant solution, but it's fast, direct and simple: drill your oversize hole, scuff up the exterior of the existing cap, epoxy in place, wait for the epoxy to cure, done.