I generated a new sitemap for my (WordPress) website with the Yoast SEO plugin.
However, Google shows the old sitemap in search results, so how long do I need to wait to see the new sitemap indexed?
Sitemaps are not indexed by Google per se. Instead, they inform the Google spider what pages, posts, tags, categories, authors, etc. are accessible to the spider and Google will then change its behavior accordingly.
Google shouldn't show a sitemap in the search results, though (at least not an XML-based one). If your previous sitemap was an HTML page or something else that Google would treat as a document and it is still publicly accessible, expect it to continue to show up until you remove it.
What you want to do is go add and verify your site's name and url then submit your new sitemap to Google.
To do so:
After doing this I have seen sites start showing the update results within 30 minutes on the search engines.