I waited 2 weeks and applied. Within 3 weeks after that, I received my SSN. I have read some posts of where some TN'ers applied on the day they received their TN and it took many months before they received their SSN. These people speculated that because their Immigration status was not immediately verifiable, that their application was put into the "I-will-get-to-it-when-I-feel-like-it" pile.
I guess my point here is that the easier you can make it for the government personnel to process your request, the more likely it will succeed the first time.
Good luck,
Al