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Make sure you have saffron crocus bulbs, which bloom in fall. Regular crocus bulbs bloom in spring. Since yours is trying to grow, they should be the right bulbs.

Here is more:

https://www.gardeningknowhow.com/edible/herbs/saffron/growing-saffron-crocus.htm

You will want to keep them uncovered if you want them to survive. Likely, they will not survive being covered for extended periods.

However, if you cannot get the supplies moved, elsewhere, you can dig up a patch to preserve in container until the work is done. Just be sure to get a very large chunk of soil so that you do not disturb the roots of the patch, at all.

https://www.gardeningknowhow.com/ornamental/bulbs/crocus

As long as the bulb is not cut, they should be OK. However, they won't get the benefit of the leaves collecting energy for next year's blooms as it sounds as if the leaves left with the blossom. So, they may be slow to recover. It could be a couple years before you get blooms again.

The mulch will not burn anything. I would point out your bulbs to the lawn service, though, to make sure they don't remove any more foliage or cover the bulbs with mulch.

https://www.gardeningknowhow.com/ornamental/bulbs/crocus/growing-crocus.htm

You may have trouble finding them for sale right now. They will probably be all mixed up when it comes to bloom time. You may get some blooms this summer if they get enough chill time after they are planted. If they don't bloom this summer, they should bloom next year.

Just be sure when planting, you don't disturb the roots of the already planted corms. Here is more:

https://www.purdue.edu/hla/sites/yardandgarden/didnt-get-your-bulbs-planted/

https://www.gardeningknowhow.com/ornamental/bulbs/crocus/when-to-plant-crocus.htm