Recently, I've learned about git remote helpers feature in Git, which allows to transparently use Git with other version control systems. As Weblate currently supports only Git, it was quite obvious to give this approach a try.
After some testing, it actually worked just fine - everything works as expected and you can use Weblate with Mercurial or Bazaar with these helpers. Of course there might be some rough edges, but all standard things I've tried worked just fine.
This is now also covered in Weblate's FAQ, which includes basic instructions on how to setup this.