Length: Up to 14.3 km (8.9 miles). Toughness: 2/10
10:21 Chessington South train from Waterloo (Vauxhall 10:25, Clapham Jct 10:30, etc), arriving Chessington South (TfL Zone 6) at 10:55. Buy a Zones 1-6 Travelcard or use Oyster PAYG / contactless.
Trains back from Cheam (Zone 5) to Victoria are half-hourly at xx:08 & xx:38.
This walk takes in some pleasant green spaces squeezed into the urban sprawl of south-west london, including the Hogsmill Nature Reserve in the morning and the historically interesting Nonsuch Park in the afternoon. I'm told the park will be muddy so I'd be tempted to ignore the rather serpentine official route and stick to the main avenue. If you fork left halfway along you can rejoin the route at the Mansion, where it's worth popping into the Nonsuch Palace Gallery (which is only open on Sundays, between noon and 4pm; free entry) to see the elaborate model of Henry Ⅷ's palace.
The suggested lunch stop is in Ewell, halfway through the walk. The route passes the Spring Tavern, while if you venture into the village you'll find the Famous Green Man. There's a café in the Nonsuch Mansion and more tea places at the end of the walk in Cheam.
The L=swc.352 page contains details of the historical features en route as well as the directions.