It was actually officially answered here back during a 'Letter From the Producer Live' video, and yes, it is indeed only a span of several days to a week or two at most between the ill-fated banquet in Ul'dah and the Warrior of Light being granted entry into Ishgard:

0:05:06
Q: Story-wise how much time has passed since the end of Patch 2.55 and the beginning of Heavensward?
A: While there is no specific mention of how much time has passed in-game, it’s about a few days to a few weeks after the events that took place during Patch 2.55. You can think of this time as the time that passed while Haurchefant was speaking with the people of House Fortemps, and the expansion begins when the Warrior of Light’s entry to Ishgard has been approved, so it’s not like months have passed.
So there you have it. There is no specific period of time mentioned that has passed, but it's around a few days to a few weeks.