While not an ideal time to repot, I don't think any real damage will be done.
Remove the plants from their soil now, get as much of the old and new soil off as you can, and set aside until the new mix arrives. This will give the roots a chance to dry out on the chance that they are staying wet too long. Succulents CAN survive out of soil for a while.
In the meantime, I would pick up some perlite while you're waiting.

Mix the perlite and the cactus soil at a minimum 50/50 ratio.....go more on the perlite, if in doubt, not less. Do not water your newly potted plants for 5-7 days, then water thoroughly so that water runs out the drainage holes. Don't water again until the soil feels dry....you can put your finger/a pencil in the soil (a la toothpick in the cake) to check the dryness.

Hope this helps.

Nancy

Good advice from Nancy :) Far better to have good, fast drainage than a soggy, wet and therefore a cold bottom. That said, it needs to be the right type of mix and one that can retain some water and food. Most C&S will appreciate a fast draining mix, there are exceptions though for e.g. the jungle type cacti such as Xmas and Easter which are epiphytes and prefer humus and a more moisture retentive mix than regular C&S.

Gill UK

Thank you, Nancy and Gill! This is very helpful. The plants are mostly echevaria and jade. I will let you know how it goes once I've repotted in the next few days! Many thanks.