According to SW Trains emergency page they won't be running a service through WP today so your only chance is to go to New Malden for the 4-an-hour Waterloo to Waterloo trains round the Kingston/Twickenham loop or see if there's enough buses to get to Morden and get the Northern Line. I suspect that's where I'll be.
EDIT: The 93 bus was starting from Stonecot Hill and not going through North Cheam this morning. The 293 will take you from North Cheam to Morden, albeit using a very circuitous route.
UPDATE: The SW Trains website has more info on their service stopping at New Malden.
London Waterloo circular via Wimbledon, Kingston, Twickenham
Services will run from London Waterloo at 12, 27, 42 & 57 minutes past each hour via Wimbledon, Kingston and Twickenham
Services will run from London Waterloo at 03, 18, 33 & 48 minutes past each hour via Twickenham, Kingston and Wimbledon
Although the bus/tube combination worked OK this morning I think I'll try the train/bus combination on the way home.