An $L(d,1)$-labeling of a graph $G$ is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least $d$ and those at a distance of two receive labels that differ by at least one, where $d\geq 1$. Let $λ^d_1 (G)$ denote the least $λ$ such that $G$ admits an $L(d,1)$-labeling using labels from $\{0,1,\ldots , λ\}$. We prove that $λ^d_1(X)\leq 2d+2$ for certain direct graph bundle $X= C_m\times^{σ_\ell} C_n$ and certain Cartesian graph bundle $X= C_m\Box^{σ_\ell} C_n$, where $σ_\ell$ is a cyclic $\ell$-shift, with equality if $1\leq d\leq 4$.