No, Britain uses a different term for Asian people than we do. Our “Asian” is their “Oriental”. Our “Middle Eastern” is their “Asian”. It is not like they are trying to hide under the term. When British people see the term “Asian”, they clearly know this means people from the Middle East. When they see the term “Oriental”, they know this means people from Japan, China, Korea, & etc. Their words are just different from ours. It is not like they are lumping Middle Easterners in with people from east Asia. They just use different terms for the two groups of people than we do.
Asian in Britain actually has two meanings.
‘South Asian’ means Indians, Pakistanis, Bangladeshis, Goans. IE those from ‘the Indian subcontinent’.
Asian without the south prefix means oriental/Far Eastern races ie Hong Kong Chinese, Thais, Koreans, Japanese etc.
The one US meaning that somewhat mystifies us here is that you regard Pakistanis as Middle Eastern.