Exact analytical calculations of eigenvalues and eigenstates are presented for quantum coupled double-well (DW) systems with Razavy's hyperbolic potential. With the use of four kinds of initial wavepackets, we have calculated the tunneling period $T$ and the orthogonality time $τ$ which signifies a time interval for an initial state to evolve to its orthogonal state. We discuss the coupling dependence of $T$ and $τ$, and the relation between $τ$ and the concurrence $C$ which is a typical measure of the entanglement in two qubits. Our calculations have shown that it is not clear whether the speed of quantum evolution may be measured by $T$ or $τ$ and that the evolution speed measured by $τ$ (or $T$) is not necessarily increased with increasing $C$. This is in contrast with the earlier study [V. Giovannetti, S. Lloyd and L. Maccone, Europhys. Lett. {\bf 62} (2003) 615] which pointed out that the evolution speed measured by $τ$ is enhanced by the entanglement in the two-level model.