  We study the atomic and electronic structures of the commensurate double
moir\'{e} superlattices in fully relaxed twisted bilayer graphene (TBG) nearly
aligned with the hexagonal boron nitride (BN). The single-particle effective
Hamiltonian ($\hat{H}^0$) taking into account the relaxation effect and the
full moir\'{e} Hamiltonian introduced by BN has been built for TBG/BN. The
mean-field (MF) band structures of the self-consistent Hartree-Fock (SCHF)
ground states at different number ($\nu$) of filled flat bands relative to the
charge neutrality point (CNP) are obtained based on $\hat{H}^0$ in the
plane-wave-like basis. The single-particle flat bands in TBG/BN become
separated by the opened gap at CNP due to the symmetry breaking in $\hat{H}^0$.
We find that the broken $C_2$ symmetry in $\hat{H}^0$ mainly originates from
the intralayer inversion-asymmetric structural deformation in the graphene
layer adjacent to BN, which introduces spatially non-uniform modifications of
the intralayer Hamiltonian. The gapped flat bands have finite Chern numbers.
For TBG/BN with the magic twist angle, the SCHF ground states with $|\nu|$ =
1-3 are all insulating with narrow MF gaps. When the flat conduction bands are
filled, the gap at $\nu$ = 1 is smaller than that at $\nu$ = 3, suggesting that
the nontrivial topological properties associated with the flat Chern bands are
more likely to be observed at $\nu = 3$. This is similar for negative $\nu$
with empty valence bands. The dependence of the electronic structure of TBG/BN
on positive $\nu$ is roughly consistent with recent experimental observations.
