How fast can capsular contracture develop after breast augmentation?

Contracture is not going to present this early. It doesn't meant that it won't present at all, nor does it mean that if the pocket is tight there is a higher incidence of it presenting. All it means is that your implant is larger than the pocket that was made for it so it feels tight. In time, your implants will stretch out the inner tissues and make room. Then you should feel them soften (that can take up to about 3 months).
In some patients this is necessary particularly if the size difference is really big (going from a AA to a D for example). Those patients are all going to feel the pocket tight. It is also possible that the pocket needs some time to stretch because it's smaller than the implant (even if your growth isn't as drastic as the cup sizes mentioned above).
In either case, it has nothing to do with the bandaid situation. Your nipple sensitivity and tight pocket sensation are both likely due to the same source. The implants are large and they are stretching your nerves; that is irritating them some. In time, everything should adjust and get better. You should keep your doctor very informed of this problem because there are some measures that could help make the process resolve a bit faster.

Congratulations on your procedure. Capsular contracture typically takes several months to start forming. One breast will start to feel more firm than the other. In severe cases the breast with the contracture can become painful and appear higher and tighter than the other side. It's important to remember that breast implants take a full six months to settle fully. Over the next several months your breast should become more round and full. If you have concerns, reach out your plastic surgeon for specific answers.

It’s not unusual for patients to experience significant
firmness and tightness following breast augmentation surgery.In the vast majority of cases, this tightness
is related to muscle spasm and swelling.This is usually a temporary problem that tends to resolve with the
passage of time.In
contrast, capsular contracture is a progressive deformity that gets worse with
the passage of time.In addition, this
problem takes time to develop.In most
cases, capsular contractures don’t develop for at least 2 to 3 months following
surgery.For
these reasons, it’s important to give your breasts time to settle.As swelling decreases and muscles relax, your
breasts will drop into a more normal position.When this happens, you may ultimately be happy with the results of your
procedure.