If they're inground, and you've had rain, they may well have started into root production already.

So long as their new location is ready, and you can move them with a rootball, or at least keep their new roots cool/damp while transplanting them, I'd do it now so they settle. New bulbs can wait as they are probably still dormant.

Sympathy for your upheaval and I hope the hard work is well rewarded next growing season.

Do it now, definitely. And yes, they should bloom next spring because they have already stored everything they need to bloom next year. But don't wait because then they will start their fall root growth.

This is September...the weather has still many warm days left and, if you notice, the bulbs are not in the stores yet.
WHY? Because its too early, the soil temperature is still much too high to plant daffs, tulips et al.

Wait.....until the soil temperature is lower; when bulbs are in the stores and the time is right to plant them.

To remind you....go to that calendar hanging on your refrigerator and mark across the OCTOBER pages...

BUY BULBS, PLANT BULBS.....RE-PLANT BULBS.

This will keep the job in your mind that it has to be done.