You don't have to wait; you can actively ask Google (and Bing, separately) to come re-crawl the site.
You would have to have access to the other site's Google Search Console, and that site would have to be verified in GSC. Then in the partner site, generate an XML sitemap with only the pages you want indexed. Check that site's robots.txt to ensure that you are not blocking search engines from the pages you want re-crawled and noindexed. (If you block the pages, the crawlers won't crawl the URL's and see your new meta tags.) Then go back to GSC and resubmit the new sitemap. As the site is re-crawled, search engines will take note of your new directives.
That said, it can take search engines a while to drop the content out of the index. Some of the content may be taken out right away, some may be suppressed deep into SERPs first and then taken out. Unfortunately, sometimes it's a process. After performing the above steps, if you still see your noindexed nofolowed content in SERPs, you can request that those URL's be manually (temporarily) removed through Google Search Console's "Remove URL's" section. (But let them crawl the site before you do this.)
A word on your new meta tags. Since you've set those tags to noindex, nofollow already, setting noarchive and nosnippet is redundant, because the first two make the page virtually invisible to search engines, and are thus sufficient.
One final bit of advice, have a chat with the partner site's server admins re: why the site is redirecting to yours. I've seen a host of SEO issues stemming from these kinds of setups. If there's no good reason why that site links to or redirects to your content with pages that shouldn't be indexed, you may want to discuss additional actions they could take, and ways to avoid this sort of thing in the future.