You can move bulbs in growth if you are careful. Dig them up and then either immediately replant or pot up but ensure you water copiously.

I would concentrate on the bulbs. If you do try to get some poppies, I would replant them right away, the bulbs could wait a day or two, or even until next weekend if necessary, in total shade preferably so they don't get too dry. Doubtful poppies would survive tilling though. Mature bulbs are tough, as you've seen - they were tilled, sprayed and now about to bloom. Wow. Anything worth the effort to you is all that might be saved, no contest really. Just depends on your time/energy. At least your neighbor is being neighborly about it, that's cool.

Thank you both for the advice and encouragement...I now have several buckets and am thinking of cutting the tulip flowers for vases, and letting the greens die back. Perhaps that will give them enough energy for next year?

I think the removing the flowers and leaving the leaves to die naturally is probably a good idea. Bulb producers remove the flowers.