The easiest way to get to Ostuni is to fly straight to Brindisi (the nearest airport) or Bari.
There are direct flights to both Bari and Brindisi from London Gatwick and London Stansted.
Alternatively you can fly to Rome, Milan, Naples or Pescara where you can take an internal flight or train direction Brindisi or Bari.
Ryanair offers daily flights from London Stansted to Brindisi and Bari. Time change depending of the time of the year.
British Airways has direct flies to Bari from Gatwick. The service runs 3 times a week Tuesday, Thursday and Saturday from March until October.
From Rome, Milan or many other places you can get a fast train to Ostuni or Brindisi.
There are frequent trains during the day and usually it does not take long to get here.
Please note that some trains called Eurostar have an extra charge. Ensure you have the right ticket before boarding.