As a sysadmin, Snowden basically had root, and probably had access to pretty much everything that wasn't thoroughly airgapped. However, very few computing resources would have been cleared for that Exceptionally Compartmented Information. The documents he gathered were focused more on activities like mass surveillance and standard undermining, that he sought to blow the whistle on, rather than their targeted cryptanalytic capabilities in general.
640-bit RSA could be broken essentially in real-time by the computing resources available to GCHQ a couple years ago. Of course, they don't actually have to work in real-time, so I suspect that 1024-bit RSA is entirely within their capabilities currently, given that. Diffie-Hellman is slightly harder, but if they're prepared to throw some in the bin or lag behind, they can probably do it, but that's just guesswork.