Is there a way to get a list of tags that the user has set up as "Watched Tags"? If not, is it on the roadmap?
No, this information is not exposed via the API (and it's not available in SEDE either). The same question was posted here on Meta Stack Exchange.
User @rene found a way to scrape it from the website; see here for his solution. It could be that the rename / new functionality (tag favorites -> tag watching) broke it, I haven't tested it.