Crawling
Automated crawlers follow links between pages and revisit known URLs. A new page that is isolated or rarely crawled may take longer to be discovered.
Indexing
Discovery does not guarantee that the source page will be indexed. Search engines decide what content to index and may crawl pages without keeping them in the searchable index.
Search Console
Google Search Console can show linking sites and pages, but it is not designed to be a complete real-time list of every backlink. Reporting can lag and aggregate data.
Third-party crawlers
Ahrefs, Majestic, Moz, Semrush, and other tools maintain their own crawlers and indexes. One tool may see a link before another, and their authority metrics are not interchangeable.
Avoid indexing guarantees
A link seller can create a crawlable page, but cannot guarantee when Google will crawl, index, value, or report the link.
Important policy context
Google's current spam policies identify buying or selling links for ranking purposes as link spam. Google also says paid advertising or sponsorship links can be legitimate when appropriately qualified with rel="nofollow" or rel="sponsored". Read Google's spam policies.