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). At the moment, your best bet is probably to scrape the tag notifications page, though this only works for your own account.
A similar question (asking for the number of users watching a certain tag, instead of the other way around) 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.