  This paper constructs homogeneous affine sandwich cellular bases of weighted
KLRW algebras in types $B$, $A^{(2)}$, $D^{(2)}$. Our construction immediately
gives homogeneous sandwich cellular bases for the finite dimensional quotients
of these algebras. Since weighted KLRW algebras generalize KLR algebras, we
also obtain bases and cellularity results for the (infinite and finite
dimensional) KLR algebras.
