bocadillo
Water Gypsy
The tube website seems to recommend Jubilee line from Stanmore/Cannon's Park to Westminster and then District line on to Fulham, this seems simple enough, is there a reason to avoid this exchange?
Most of the District line trains at Westminster will be going to either Ealing Broadway or Richmond whereas you need a Wimbledon train. If a Wimbledon train is not shown on the destination indicator, take any train to Earls Court and change to a Wimbledon train. It's a cross-platform or same platform change at Earls Court.
If a Wimbledon train is shown as the second or third train at Westminster, unless you are in a dashing hurry, you will be better to wait for it to avoid the inevitable congestion at Earls Court.