Which of the following compounds is the most nucleophilic in polar protic solvents?

The answer is B. The question might as well be restated: "which is the most basic in water?" since a nucleophilic describes the behavior of a base. also water is a polar protic solvent.

C & D can be eliminated off the bat because their structures are similar to A & B. They will have similar properties as their counterparts, but be weaker in them since their atomic radii is bigger and bond overlaps weaker.

Hopefully at this point, it is common sense that water is a neutral compound and NH3 a [Lewis] base. but if you want to know why NH3 is a stronger base, it's cause of the symmetry. if it takes one more H+, it becomes symmetrical and the energy is dispersed better. water would need to take 2 H+, and carry an unfavorable charge to do the same.