Probably near a million. There are 200 realms in North America alone, and each realm is capped at a few thousand players. If we assume that, at peak login (say, the 24 hours after the last expansion pack was released) there were an average of 2.5k people online per realm, that would be 500k in North America alone...I believe I remember reading that 50% of WoW players are in China, and there's also Europe, Russia, South America, etc. to consider.
From another angle, with 12m players worldwide, I think it's safe to assume that at some point in the week after an expansion launches, 1 in 12 players is online.
That's a lot of cores. While the hardware and interconnects are certainly different, a number for perspective:
The Jaguar supercomputer at Oak Ridge National Laboratory, currently the most powerful supercomputer in the world, has 224,256 compute cores.
http://www.nccs.gov/computing-resources/jaguar/
If all of the WoW players woke up tomorrow and realized they didn't want to play a silly game anymore, Blizzard could make one heck of a distributed computing cluster.
This post seems to interchange between referencing WoW and Blizzard's "gaming operations" a few times. I assume gaming operations would include Battle.net.
Is the data here strictly about WoW? I'm just curious...
EDIT: The linked Gamasutra article clarified it. These stats are for WoW only.
Yes, they recently required that all accounts switch over to Battle.net (the cutoff date was sometime early november, after a months notice).
The public reasoning about it is so users can use the same email/password combination for WoW as well as the other upcoming Blizzard games like Starcraft 2, Diablo 3, and whatever their new IP next-gen MMO ends up becoming. The private reasons one can only guess, but I would assume at least that they can do stat tracking much easier with the new system.