Pakistan has never been ruled by the British or anyone else for that matter. So you couldn't have a British Pakistani passport.
Joining an indian political party doesn't make you indian. priti patel is a cabinet member of the British Conservative Party but she is NOT ethnically or racially English or white western European.
If you joined the Gambian Labours Workers Party, that wouldn't make you Gambian or Black African.
They were citizens of the "British Raj". Nothing more. Here is their flag and country:
View attachment 714980
Even Myanmar was a part of the" British Raj" yet they were never considered indians.