They discussed this in the approval meeting. There was only one case ever in any vaccine or trial of a side effect showing up after 6 weeks, it was over 50 years ago in a live virus polio vaccine on an immunocompromised person.
Given the many billions of vaccines given of various types, it is about as close to certain as we get in this world to not expecting long term side effects ever.
The reason they aren't concerned about it is for good reason, not because they "haven't looked into it". You have to look at relative risk.
This decision to lift the total ban on this vaccine was probably the easiest decision the FDA ever had to make. This one isn't even a question.
Now they just have to lift the ban on the rest. The standard for EUA is supposed to be "net expected benefit", which they are not following. They are currently only approving EUA when they hit "no conceivable downside", which is a too high standard.