Confusion Regarding Nitrogen Inversion In nitrogen inversion, the lone pair changes its direction continuously. 

Does this in any way delocalize the charge density on the nitrogen making it less nucleophilic and less basic than another nitrogen attached to, say, a bridgehead carbon?

Consider the specific case of 
Quinuclidine and tri-methyl amine. Which will be more nucleophilic among them and why?
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.