You have a slight error in the first name, it's
Amaterasu 天照. The characters are 天 "heaven" and 照 "shining," so this roughly means "shining in heaven."
Uzume is short for Ama-no-uzume, which can be written 天鈿女 or 天宇受売. These are tough to translate, and since there are two distinct options, neither of which makes much sense, I have a feeling the original meaning has been lost. 天鈿女 is "heaven" + "ornamental hairpiece" (possibly referring to a crown) + "female." 天宇受売 is "heaven" + "heaven" (another character with the same meaning) + "receive" + "sell."
By Googling I found that several sites give the meaning of Uzume as "whirling," but this really does not seem reliable to me. I suspect that this arose from confusion with the word uzumaku 渦巻 "to whirl."