Which one of the Nitrogen atoms in $\frac{{\mathrm{H}}_2 \mathrm{N}}{\mathrm{I}}-\frac{\mathrm{NH}}{\mathrm{II}}-\mathrm{CO}-\frac{{\mathrm{NH}}_2}{\mathrm{III}}$ is the most nucleophilic?

The electron rich species are called nucleophiles having high electron density. These species tend to attack the electron deficient site also known as the electrophilic center.

From the given compound, two amino groups attached to the $\mathrm{C}=\mathrm{O}$ group so, they participate in conjugation. Hence, the electron density over II and III are less as compared to the first ${\mathrm{NH}}_2$ group as the electron density over it is intact because they do not participate in conjugation.

Hence, I is the most nucleophilic nitrogen atom.

Therefore, option $\left(B\right)$ is correct.