No, I don't think Cheryl is of mixed race. Her father looks very much like he origin from England, the same as her mother. I think the reason why you think that is because they both have dark hair.
My sister is from Norway (same as me) but she has dark hair and a little darker skin, so a lot of people think she comes fra Iraq and everything. A little off topic, but my point is that even though they look Asian doesn't mean they are.