The recommended way of travel from Heathrow or Gatwick airports is via the Oxford Airline bus. Tickets can be purchased in advance online. Direct buses depart regularly from Heathrow's Central Bus Station (serving terminals 1,2,3) and Stand 10 at Terminal 5. A free shuttle bus connects to Terminal 4. Get off at Oxford's Gloucester Green bus station (the final stop). From Oxford, Bus X2 departs from Park End Street for Abingdon; alternatively Bus X1 and Bus X3 depart from St. Aldate's. Abingdon is a small town and it will be a fairly short walk to Cosener's House (see map).
Alternatively a train can be taken from Heathrow rail station at Terminal 5 to Didcot Parkway railway station. Bus X2 runs from the station to Abingdon.