Let N₁ denote the restricted nullcone of the Lie algebra g of a simple algebraic group in characteristic p>0, i.e. the set of x∈g such that x^|p| = 0. For representatives e₁,...,e_n of the nilpotent orbits of g we find the irreducible components of g^ei∩N₁ for g = G₂ and F₄ in good characteristic p. We do the same for g = E₆ with the exception of three nilpotent orbits. We use this information to determine the irreducible components of the restricted nilpotent commuting variety C₁^nil(g)= {(x,y) ∈ N₁×N₁ : [x,y] = 0} for g = G₂ and F₄. We do the same for g = E₆ with the exception of when p=7 where we describe C₁^nil(g) as the union of an irreducible set of dimension 78 and one of dimension 76 which may or may not be an irreducible component.