How does gaussian convution stacking work? If I first convolve an image with a Gaussian filter with σ = 1, and then convolve
the output of that image with a Gaussian with σ = 2

Does this give the same effect as convolving the original image with σ = 3?
From this Wikipedia page, you can get:



So for your example, $\sigma_X = 1$ and $\sigma_Y = 2$. The result has a standard deviation of $\sigma_Z = \sqrt{\sigma^2_X + \sigma^2_Y} = \sqrt{1 + 4} = \sqrt{5}$.