Believe it or not New Malden is the preferred home for the Korean community in London and the first things you will see on leaving the station are a couple of Korean restaurants! We will walk via back streets and paths to Wimbledon Common. Then we will walk through the middle of Caesar's Camp, an ancient British fortification, skirt Wimbledon village and cross the common to the Windmill. Lunch can be had here in the cafe (or bring your own food). Public toilets. The mill may be open for visits. Then we continue across the common and go under the A3 to reach Putney Heath. At this point anyone who has had enough can catch a bus into Putney for a train. Otherwise we will continue via Putney Park Lane (an odd unsurfaced road that was probably once the driveway of a grand house). At Upper Richmond road there is another chance to catch a bus into Putney. From here it is only a few hundred yards to Barnes Common. We will pass Barnes station, from where there are frequent trains to Vauxhall. Continuing across the common and through the village, with its pond, we reach the Thames. There is then the option to catch a train from Barnes Bridge station to Vauxhall (or a bus to Hammersmith) or turn right and continue on the Thames path to Hammersmith for the Piccadilly Line home. Mileage is about 5.5 to Barnes (less if you finish at Putney Heath), 6.5 to Barnes Bridge or 8.5 to Hammersmith. 

I know this is south of the Thames, but you won't meet any dragons. Look out for Wombles though! It is possible you may meet a bit of mud in places if it continues to be wet, but generally it shouldn't be bad underfoot.