You still haven't explained. I understand perfectly that the countries you mention were part of the British Empire and most Brits would be more familiar with people from those 3 countries, but it still doesn't answer the question:
Since the majority of Indians are Hindus and not Muslims - why would you describe someone leaving a mosque as Asian instead of Muslim?
Why do American police sometimes describe someone as 'black' when they are actually of Jamaican descent?
Probably because they are speaking of the person's race - rather than their country of birth. A Jamaican could also be blond haired and blue eyed, if they are of British stock. If you are speaking of race, you are speaking of color - not national origin. If the police are doing a manhunt and described the person as a 5' 10" Jamaican - that wouldn't be a very good description. Anyone could have been born in Jamaica. If the police describe the person as a 5'10" black male - then everyone knows exactly who they are looking for.
"Since the majority of Indians are Hindus and not Muslims - why would you describe someone leaving a mosque as Asian instead of Muslim?"
Because the description 'asian' is used as an ethnic description. It is not in any way a synonym for 'muslim'.
Your paragraph on why American police tend to give the ethnic description 'black' applies equally to why British police tend to use the ethnic description 'asian'.