That is a difficult question to answer. The first thing is to throughly inspect the places where the wood is moving.

It looks as if you may have access to the underside of the stairs, if you do then you will be able to tap the wedges in as they are probably why you have movement. The stairs are cheaply finished as shown by the bandsaw marks still showing on all the treads and risers. While not proof that the original makers skipped steps in production, it certainly makes it a significant possibility. This can be both good and not so great, the good point is that tightening the treads up may be easy, the not great point is that they may have used nails to avoid the time to make and fit the staircase properly.

TLDR just renailing will work for a short time it is a bodge specially if you don’t use cut nails.