Running UWP apps is possible, at least with some apps. I've successfully run the Trello UWP app (which is only available from the Microsoft Store).
First, you need to acquire the .appx file for the UWP app. One way to do this without Windows or the Microsoft Store is to use the this site:
Paste the Microsoft Store URL of the app into the site. For example, the Trello app's URL is:
This gives you links to all of the files available through the Microsoft Store. You only need the .appx file, which may be available as multiple versions (e.g. x86 and x64).
Chrome doesn't want to download the files simply by clicking the links. This made the site feel a bit sketchy at first. In Chrome's DevTools console, Chrome says that it's not downloading the file, because the Microsoft Store URL uses http://, and the site uses https://. But since the file is coming straight from microsoft.com, it feels safe to me.
Right-clicking the link, selecting "Save link as", and choosing "Keep" allows you to download the file despite Chrome's objections.
Once you have the .appx file, you can extract it. It's just a regular zip file, so run something like
unzip -d output-dir file.appx or
atool -x file.appx.
The extracted files should have a directory called "app", which should have the executable and other files for the app.
For the Trello app, just running
wine Trello.exe inside the "app" directory worked for me. But other applications might not work straight away, if something more complex happens during their installation.