My newborn broke her humerus mid-shaft during birth. there is no cast or splint on it, will it heal straight? how long will it take?

Yes: Even if there is a slight curve to the bone now, the bone will straighten as it lengthens as she grows. The approx healing time is 4-6 weeks.

No time at all: Injuries sustained during birth start healing right away. Your daughter should stop hurting within a couple days and the bone should be fairly solid by two weeks. No cast is required; i usually treat humeral shaft fractures in this age group by pinning the sleeve of the onesie to the body of the onesie. Congratulations and don't worry. In a few months it should be as good as new.