"British" black tea is predominantly Sri Lankan or Indian blends. If your Asian stores are East Asian (Chinese/ Vietnamese) they are unlikely to have it.
In China, what we call black tea is known as red tea! 红茶 (simplified Chinese as used on the mainland) or 紅茶 (traditional Chinese as used in Hong Kong and by much of the Chinese diaspora).
What the Chinese call black tea (黑茶) is something else - fermented teas such as Pu-er. Very different from "British" black tea.