I haven't been able to get a direct answer online. Can a woman get HPV/genital warts from a partner who has cold sores during oral sex?

Different virus. The cold sores are due to Herpes simplex virus (HSV) type 1 or 2, while the genital warts are caused by strains of the human papilloma virus (HPV). Oral sex with a cold sore may transfer that virus, but not the other. If the person in question also has or has had genital warts, they may be shedding that virus and pass it during genital to genital contact.