Here is an easy counterexample under the Continuum Hypothesis, where the <em>union of the topologies</em> generates a non-normal topology.
Take the upper half plane $\mathbb{H}$ (including $x$-axis) and enumerate $\mathbb{R}$ as $\langle x_\alpha:\alpha<\omega_1\rangle$.
Now let topology $\mathcal{T}_\alpha$ be generated by the normal topology plus the tangent disk neighbourhoods at all points $(x_\beta,0)$ for $\beta\le\alpha$.
Each topology $\mathcal{T}_\alpha$ is still separable and metrizable (regular with a countable base) but the union generates the Tangent Disk topology, which is not (even) normal.