It turns out that both distributions have a sort of gauge-invariance which makes the question irrelevant; use other hints to make the choice between Gaussian and gamma.
First, let's walk through the Gaussian case, which is much simpler. Let $\mu$ and $\sigma$ be our mean and standard deviation respectively, and note that they carry the same units. Given a scalar $k$ which maps from one (univariate) gauge to another, we want $k\mu$ and $k\sigma$ to be our new mean and standard deviation with the new units. Continuing the example in the question and considering units of time, we might want to map 2min ± 1min to 120s ± 60s by setting $k=60$ and mapping from minutes to seconds.
That all follows directly and effortlessly from definitions. Now, let's work out the same scaling for gamma distributions. We'll use the alpha-beta definition, so for the distribution $\Gamma(\alpha,\beta)$, set:
$$
\mu = \frac{\alpha}{\beta} \\
\sigma = \frac{\sqrt{\alpha}}{\beta}
$$
Since $\mu$ and $\sigma$ have the same units, by dimensional analysis, $\alpha$ must be dimensionless and $\beta$ has the inverse units of $\mu$. We should therefore hope that scaling by $k$ means no change to $\alpha$ and an inverse multiplication to $\beta$, so that the units work out. And indeed, we can massage our equations after scaling:
$$
k\mu = \alpha\frac{k}{\beta} \\
k\sigma = \sqrt{\alpha}\frac{k}{\beta}
$$
Which is $\Gamma(\alpha, \frac{\beta}{k})$. So gamma distributions are not clueless about the choice of gauge, but it is straightforward to rescale the distribution for any particular change in gauge.
