That the extremely earnest new tuner “White Rose: The Musical,” now at Theatre Row, works as well as it does is a testament to a very talented cast who handle their roles with sincerity and seriousness, as well as a pleasing-enough score by newcomers Brian Belding (who also wrote the book) and Natalie Brice, and the simple yet effective direction of Will Nunziata. And, at 95 minutes, this poignant yet slender tale doesn’t wear out its welcome.