It is well known that if $f:[0,1]\to\mathbb{R}$ is continuous then
$$ \begin{align} B_n(x) &=\sum_{\nu=0}^n f(\nu/n) \binom{n}{\nu}x^\nu (1-x)^{n-\nu} \end{align} $$ converges uniformly to $f$ as $n\to\infty$, and that (see e.g. Bojanic & Cheng 1987)
$$\| f-B_n \|_{\infty} \le \mathcal{O}(n^{-1/2}) $$
where the constant depends on the modulus of continuity of $f$.
There is a related result for functions that are defined on the positive real line: if $f:[0,\infty)\to\mathbb{R}$ is continuous with $\lim_{x\to\infty} f(x)=0$, and
$$S_{u,n}(x) = e^{-ux} \sum_{k=0}^n f\left(\frac{k}{u}\right) \frac{(ux)^k}{k!},$$
then $\lim_{u\to\infty} \lim_{n\to\infty} S_{u,n}(x) = f(x)$ uniformly for $x\in[0,\infty)$. See e.g. Devore & Lorentz.
What is the rate of convergence in this case? I'm particularly interested in estimates when both $n$ and $u$ are finite.