How fast does a bipolar child cycle between mania and depression?

Also can be mixed. As mentioned, "cyclicality" can vary greatly. In addition, episodes can be "mixed", with suymptoms of both mania and depression manifesting concurrently.
Varies. Do see a well trained child psychiatrist.
Mood Disorders. What was previously referred to as Bipolar Disorder in children is now called disruptive mood dysregulation disorder (DMDD) under DSM V. Children can cycle very rapidly (or some may have longer durations in the different phases). Care by a child psychiatrist is very important.