The Hare Krishnas were formed in the 60s in New York, which doesn't exactly give them a lot of legitimacy. I think the best bet is to have them all go somewhere to form a Jonestown-style settlement (though preferably without the final stage...)
Happily, OTL there's a strong push to form self-sufficient societies in the Hare Krishna, since getting getting back to the Earth and living a simple life in harmony with nature and stuff like that is part of their doctrine.
It seems easy to me to imagine large numbers of Hare Krishnas moving to some remote location in a poor country and setting up their own independent settlements, and maybe even getting a few tens of thousands of people there. Getting them in charge of a country is hard, though, since they're pretty pacificistic.
Maybe have them set up in one of the African states that gained independence late (Namibia? Somaliland?) and have their community be large enough that the colonial masters try to chop them off as their own small country.
Having an autonomous region within another country would be easier, though, and carries the advantage that it could probably be done somewhere more likely to allow it. In both cases in Africa, it seems likely that someone would try to end the experiment, either from desire for wealth or from anti-European sentiment.