Prune out dead branches or portions of twigs that are dead. Wait until late winter or early spring, after the threat of frost passes. Newly emerging leafy buds reveal which branches, or where on branches, living tissues remain. This helps guide how much trimming is really needed.
Make pruning cuts with a bypass or hand pruners to remove the dead branches and twigs. The pruning cut needs to be 1/4 to 1/2 inch away from the alive dormant bud, emerging new leaf, or extant leaf that survived the winter.
Stagger the cuts across the canopy of the weeping bottlebrush. Trimming off old leaves, make pruning cuts at varying lengths on the many weeping branches to help mask the pruning maintenance. Avoid cutting branches back to the same height or general location on the plant's branch canopy, as it will accentuate the fact you pruned.
Balance out the trimming across the bottlebrush's weeping canopy if you find one side of the plant sustained more winter damage than the other. The bottlebrush will respond favorably to the light trimming by late spring, sprouting new replacement shoots that fill-in the weeping silhouette of the canopy. Staggering the cut heights, but balancing the trimming on all sides, creates a uniform canopy that will become fuller and green by the end of spring.