They will not sprout again, but you may have a problem with Honey fungus if you leave the stumps in the ground.
Having said that we have 4 stumps in our front garden and they have been there for 10 years without any problems.

It's probably too late for you to do this now, but I decided to get rid of a Leylandii that was too near my conservatory. I chopped it off about 6 feet from the bottom, removed the branches from the remaining trunk and screwed a hanging basket support to the trunk.

We had a row of 15 leylandii trees that had reached about 40', and were shading too much of the garden, taking too many nutrients and hitting power lines. We cut them down 3 years ago and the trunks haven't sprouted at all. In fact, they make a splendid row of 4' high fence posts which, with plastic covered mesh nailed between, keep wildlife away from the vegetable garden very effectively. No sign of honey fungus, but ivy has grown up some of them very picturesquely.