That depends on your definition of site link.
In Google’s world, a “site link” is a specific term that refers to additional links that show up under main organic listings. (See screenshot above.) Google [thoughtfully] selects these links for you based on a variety of factors including what they think is most informative or useful to users. As a web site owner, the best you can do is request the removal of the ones you don’t like. You can’t pick links you like.
If the question is about links to your site from other sites, then the answer depends on when those other sites are crawled.
Google won’t find inbound links from other sites to your site by crawling your site. They find inbound links by crawling the other web site – the one that links to yours – and then storing that data in its index. So if the site that links to yours is really popular, then your link might get picked up quickly. It could also take days.
Keep in mind that not all sites are crawled at the same frequency. Some sites are crawled more often because they’re more authoritative or they post more frequently or they themselves have more inbound links. It’s also helpful to know that not all of the URLs on a given web site are crawled at the same rate. Often, pages that are updated frequently get crawled more than pages that have a more evergreen style kind of content. And the only way you can know that for certain is to look at the other site’s log files (which will show you how many times a day ever page on the site has been visited over time) and making an educated guess based on past history.
Of course, it’s not likely that the other site owner will let you see their log file data so the real answer to the question is, “It’s up to Google.”