We do the exact same thing when moving a tree to a new location. Cutting the roots helps a tree to produce new feeder roots within a rootball that is manageable. Very basic way to move any plant, shrub or tree. Just don't do it again. And water often and deeply.
I'd go get some mycorrhizae at a plant store to help the roots left operate better. Mycorrhizae is a common fungus that works with roots in a symbiotic way. Whenever I transplanted any tree, shrub, I'd always apply some of this fungus. And even moving 20' Rhododendrons in the middle of summer I've not had a single death yet.
Do allow the soil to dry out before watering deeply again. You could also get some 3 or 4 inch PVC, drill holes every 6" beginning with the rim. Insert it at an angle within the drip line (draw a line to the ground from the perimeter of the canopy, make a circle around the base of your tree. Within that circle is an upside down version of the canopy. Not as deep as the depth of the canopy, roots are shallow. When you water, also put water in the pipe. Roots are pretty much within the top 6" of soil and not much farther. This will encourage deeper root growth to get at the moisture down deeper. When the soil dries out at the top 2 inches of soil, water deeply again. Forget fertilizer or any other additives until spring.
Something else you should do is lightly prune by thinning not heading. Reduce the canopy that needs lots of water. This will lessen stress of your tree. Need to reduce canopy when you cut roots.
Why are you tilling?