Young walls and graded dimension formulas for finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$

We study graded dimension formulas for finite quiver Hecke algebras $R^{\Lambda_0}(\beta)$ of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluating at $q=1$, the graded dimension formulas recover the dimension formulas for $R^{\Lambda_0}(\beta)$ described in terms of standard tableaux of strict partitions.