Typically this arrangement does give you a higher allowance. The two things to look out for are
1) what it says on the e-ticket - that's your contract and it's what the agent will see on their screens.
2) whether the Vistara flights are codeshared or primed. Did you buy them with a BA flight number?
Cabin luggage does follow operational rules for each airline, so I suspect BA being more generous there. The other main wrinkle is that status benefits, if you have status, won't carry over to non oneworld airlines, including Vistara. And within oneworld there are some inconsistencies.

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