Yes, amine inversion does effect(reduce) basicity and nucleophilic character of amines
In amine inversion a nitrogen which is sp3(tetrahedral) converts into sp2(planar) and then again coverts into sp3(tetrahedral).
This happens because nitrogen has a lone pair and this lone pair moves through nucleus towards the opposite side. This is known as quantum tunneling. While the electron pair is moving through the nucleus nitrogen has sp2 hybridisation. So, if nitrogen inversion is effective availability of lone pair decreases.
Now let's come to the case of quanidine. In quanidine, nitrogen inversion doesn't happen occur because of the impossible geometry it would have to form to do so.
However, in tri-methyl amine there is no such hindrance and hence achieve planar state relatively easily.