It is still possible to download the game through non-Uplay services such as Steam or Origin, but all versions require Uplay to connect in order to function.
As usual, the widely leaked pirated version suffers from none of these issues and can be played offline right now.
Here's the response from the official Watch Dogs Twitter account:
We are experiencing issues with the authentication services. Players may experience long delays when trying to login in-game. More to come.
As usual, check out our Watch Dogs article for up-to-date bugs and fixes, which we'll be sure to investigate further as soon as we can login to the damn thing.
Edit 10pm 27/05/14: Uplay servers appear to be up and working, can now login and play Watch Dogs.