We present results of the first lattice QCD calculations of $Bc\to Bs$ and $Bc\to Bd$ weak matrix elements. Form factors across the entire physical $q2$ range are then extracted and extrapolated to the physical-continuum limit before combining with CKM matrix elements to predict the semileptonic decay rates $\Gamma (Bc+\to Bs0\ell ―\nu \ell )=26.2(1.2)\times 109s-1$ and $\Gamma (Bc+\to B0\ell ―\nu \ell )=1.65(10)\times 109s-1$. The lattice QCD uncertainty is comparable to the CKM uncertainty here. Results are derived from correlation functions computed on MILC Collaboration gauge configurations with a range of lattice spacings including 2+1+1 flavours of dynamical sea quarks in the Highly Improved Staggered Quark (HISQ) formalism. HISQ is also used for the propagators of the valence light, strange, and charm quarks. Two different formalisms are employed for the bottom quark: non-relativistic QCD (NRQCD) and heavy-HISQ. Checking agreement between these two approaches is an important test of our strategies for heavy quarks on the lattice. From chained fits of NRQCD and heavy-HISQ data, we obtain the differential decay rates $d\Gamma /dq2$ as well as integrated values for comparison to future experimental results.