It depends on your white. Recessive white in bantam cochins covers up any color genetics the bird otherwise has, like a coat of paint. When you breed that bird to another color, the chicks only get one recessive white gene from the white, not enough for a coat of paint. Whatever else the bird was hiding gets passed on as well, and it combines with the blue's genes to create the offspring's color. If your white has blue genes, you may get splash chicks. My guess is that you will get blue and black with silver or gold leakage.
Find threads about crossing white birds to other colors; there's plenty of them, and I'm sure you'll find it explained much better than I just did.