This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they are generated by certain Atiyah-Bott generators that appear in the Chern classes of the universal bundle.

In the paper by Beauville and Laszlo:

http://www.projecteuclid.org/euclid.cmp/1104270837

they show that the stack of bundles can be seen as fiber product of (1) the stack of bundles on a formal disk and (2) the stack of bundles on the compliment of a point.

The fiber product takes place over the stack of bundles on the punctured formal disk.

Has anyone been able to compute the cohomology of the stack directly from this description?

This might involve connections with the infinite Grassmanian. As a silly example, suppose I am over the complex numbers and the curve is $\mathbb{P}^{1}$. Suppose I take line bundles of first Chern class zero. All such bundles can be trivialized on the formal disk and the complement of the point. The automorphism groups of the trivial line bundle on the formal disk and the complement of the point are just $\mathbb{C}^{\times}$ and $\mathbb{C}[[t]]^{\times}$. So the stack of line bundles of all degrees should be $[\mathbb{C}^{\times} \backslash \mathbb{C}((t))^{\times}/\mathbb{C}[[t]]^{\times}]$.

Then one can separate out $\mathbb{C}((t))^{\times}/\mathbb{C}[[t]]^{\times} \cong \mathbb{Z}$ into the orbits generated by the degree $d$ of the bundle $t^{d}\mathbb{C}[[t]]^{\times}$.

Finally, one sees $[\cdot / \mathbb{C}^{\times}]$ for each degree, so here its the correct answer. It seems odd though that here, $\mathbb{C}((t))^{\times} = \mathbb{C}((t)) - \{ 0 \}$ looks like an infinite dimensional sphere which should be contractable, although perhaps there is a different topology in play here.

Are there any methods to see cohomology classes directly from this description, even in the case of higher rank bundles on $\mathbb{P}^{1}$ with first Chern class zero? For rank $2$, there should be $2$ generators in degree $2$ and one in degree $4$ if I remember correctly, so these should be visible in the cohomology of $[GL_{2}(\mathbb{C}) \backslash GL_{2}(\mathbb{C}((t)))/GL_{2}(\mathbb{C}[[t]])]$. There should be a stratification of this stack corresponding to the bundles $\mathcal{O}(j)\oplus \mathcal{O}(-j)$ and some kind of spectral sequence starting with the cohomology of the strata and converging to the cohomology.