After jumping into Wildstar for the first time this past weekend, I can also confirm that SWTOR has nothing to worry about... to a certain extent.
Wildstar gets a lot right with its fun combat, interesting combos for class abilities made possible by the LA, Fun PVP (with ranked/cross servers/warplots/arenas in from launch), and a brutal endgame raid scene.
However I, like many others, have been ruined by SWTOR's excellent leveling content paced by the fantastic VO work. Wildstar's chat box based story, while often inducing LOL moments is quite lackluster in comparison. This will be the biggest turn off for people who try out Wildstar and it makes getting invested into the game's early moments a difficult task.
At the end of the day, I think Wildstar will appeal to a different type of crowd. Those looking for a game that supports PVP right off the bat, with a brutal endgame raiding scene (the craziness of 40-mans, attunements, 42 weeks of daily grinding to get those extra amp points etc) and an excellent housing mini-game will find that Wildstar is the game for them.
Those looking for a more casual PVE/Raid scene, a highly popular IP setting to RP with, and more "realistic" setting (in comparison to the cartoony feel of Wildstar) will find SWTOR to be the place to be.
I myself plan to keep subs up for both games for the foreseeable future, but will likely make Wildstar my main game (in pursuit of all the glory of raiding tough end game content). I'll always love SWTOR (been here since launch) but after clearing all the current PVE content, doing all 8 class stories, and defending nodes with my Sin till my eyes bled, I'm ready to move on to another game to scratch my MMO itch.