My round-leaved sundew has just come out of dormancy last month but tonight temps are supposed to go down to -1.C. What effect will that have on my sundews?

Are your sundews temperate or tropical? Do you know the species?

In either case, bring it inside for the night to be safe. That's too cold.

While it won't trigger dormancy (wrong time of year), assuming the plant is even temperate, freezing temperatures can hurt your plants, especially living in pots that can freeze easily.

I think Round-leaved sundew = drosera rotundifolia.
(When talking about carnivorous plants, we usually use scientific names to avoid ambiguity)

Considering it is found through Siberia, I doubt that one -1 C night will kill it. Maybe cause it to drop its leaves but not kill it...

Anyway, like veronis said, bring it inside. A few days with no sun won't hurt the plant.

I collected it from the wild as it grows naturally in Ireland so...
Also, what does a D.rotundifolia's dormancy bud look like? I am looking to get more.

Like alot of very small curled up leaves google is always a good idea