If the target and source are both trusted, it can be very fast. In matter of days.
However, this is rarely the case because everything can turn to spam and get corrupted so engines are suspicious of all.
Also, there are algorithms in place that will artificially hold the backlink's influence or show downward ranking, testing how the author will react. If it's a PBN link for example, author will freak out and remove the link, sending very bad signal.
3 weeks is a good average and a starting point.