How can we determine that a particular neutral molecule is a strong base or a nucleophile?

Nucleophilicity roughly parallels basicity. The trick lies within the word roughly. A base (in a BL sense) attacks protons, whereas a nucleophile attacks anything else.

From here, try to think about this: basicity is a subset of nucleophilicity. All nucleophiles are Lewis bases; they donate a lone pair of electrons. A “base” (or, better said, a Brønsted base) is just the name we give to a nucleophile when iit forms a bond to a proton (H+). Therefore, when we are talking about basicity and nucleophilicity, we are describing these two types of events.

If basicity can be described by means of equilibria, nucleophilicity can be described in terms of reaction rates. Acid-base reactions are fast equilibria.

Many reactions of nucleophiles are not reversible and two more factors must be accounted for, though, when dealing with nucleophiles: their steric hindrance and solvent effects. The more sterically hindered a nucleophile, the weaker it is. The more polar (or, even worse, the more protic) a solvent, the weaker the nucleophile.