There is no direct connection from Ilford to Putney. However, you can take the train to Stratford Underground Station, take the Tube to London Waterloo, then take the train to Putney. Alternatively, London Underground (Tube) operates a vehicle from Ilford to Putney every 10 minutes. Tickets cost $5 - $7 and the journey takes 58 min.
From Ilford train to Stratford Underground Station, Tube to London Waterloo, then train to Putney. 48 min £11. From East Ham Underground Station Tube to East Putney Underground Station. 58 min £4. From Ilford taxi to Putney. 35 min £67. From Ilford drive to Putney. 35 min £2. From Ilford Uber to Putney. 35 min £32.