Google’s John Mueller said on Reddit that if you want to slow Google from crawling your site (or speed it up), it can take about a day from when you set the crawl rate setting in Google Search Console for it to kick in. The official documentation kind if implies it takes two to three days.
John said in the thread “Set the crawl rate down in search console. That takes a day…” He then added “if it remains a problem, use the form in the help center to report a problem with googlebot.”
The documentation says “two or three days later, when Google’s crawl rate has adapted, you can remove your robots.txt blocks or stop returning error codes from step 1.” That implies it takes two to three days but maybe not.
In fact, the docs talks about “Immediate relief” options for when Googlebot is causing server issues. Even the “Immediate relief” options can take a day for Google to pick up on and adapt.
In any event, keep in mind, the fastest option is probably to increase your server capacity and resources or block Googlebot, but almost no one wants to block Googlebot.
You can learn more about the crawl rate feature in the help documents.
Forum discussion at Reddit.