How long will a broken elbow take to heal?

A few weeks. Time to heal will depend on a few factors - such as presence of dislocation or not, presence of joint involvement or not. An uncomplicated elbow fracture will take 5-6 weeks to heal.
6 weeks. Almost all fractures in adult humans take around 6 weeks to heal to about 75% tensile strength, but take another 5 months to remodel back to their native strength. Fractures into joints take longer and fractures at the middle of bones can take longer.