There aren’t any OneWorld connections to Pune unfortunately and it’s a rubbish airport.
Easiest way is to Mumbai and then take a taxi, providing you avoid the morning and evening peaks. The BA departure (which they’ve now cut down to once daily) gets in just after midnight which doesn’t really work for the early morning Jet Airways departure to Pune if you fancy a half decent nights sleep.
if you do overnight in Mumbai, try and stay to the east of the airport in one of the airport hotels as its far quicker to get to the expressway.