I have finally, after literally trying pretty much every single markdown Apple Watch app, found a way to access Obsidian notes on Watch:
The app below (steep price warning) can use a custom folder (Obsidian vault) if you unlock it via an extra 3$ in app purchase.
Then you have to go into the ‘info’ section and check “Show as task list”.
Then flag the files you want to sync as ‘due tasks’. This way they will get synced to Apple Watch. Seems like the only app that can do that out of the box: Notebooks – Write and Organize on the App Store
I got in touch with the developer and they are planning to make a complication as well as other ways to sync to watch than due tasks, e.g. whole notebooks.
I’m really happy I finally found a solution, and it doesn’t even require syncing workflows, shortcuts or similar.
In the watch app, you can even append to an Obsidian note 
I have only tried this using Obsidian sync, not iCloud sync, so not sure if it’ll work well with iCloud sync too.