Here's a simple and far from optimal condition guaranteeing unifolrm convergence. 

Suppose for  that there exists $C>0$ such that  for any positive  integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$
$$
\int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n.
$$
In this case the associated Hermite series  is
$$
\sum_{n\geq 0} \frac{c_n}{n!} H_n(x),
$$
where
$$
c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx.
$$
and this converges to $f$  uniformly on compacts.  This follows [from known asymptotic  estimates for Hermite polynomials][1].

 For more precise results you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.


  [1]: https://dlmf.nist.gov/18.15#v