  In the present paper, we generalize the theory of quantitative homogenization
for second-order elliptic systems with rapidly oscillating coefficients in
$APW^2(\mathbb{R}^d)$, which is the space of almost-periodic functions in the
sense of H. Weyl. We obtain the large scale uniform boundary Lipschitz
estimate, for both Dirichlet and Neumann problems in $C^{1,\alpha}$ domains. We
also obtain large scale uniform boundary H\"{o}lder estimates in $C^{1,\alpha}$
domains and $L^2$ Rellich estimates in Lipschitz domains.
