'Make ends meet' is the standard idiom in BE, too.
make ends meet - definition in the British English Dictionary & Thesaurus - Cambridge Dictionaries Online
(Indian English make both ends meet) to have just enough money to pay for the things that you needCoping and not copingDealing with things or peopleTolerating and enduringBudgeting and saving money (Definition of make ends meet from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press)
What is your point?
It's less common probably because you can't have one end meeting the other without the other meeting it. Making one end meet doesn't make sense.I tried to verify if 'make both ends meet' is British English before I made a google search and found the Indian English notion. Evidently the idiom with 'both' is less common, isn't it?