"It was probably in '44. Woody was a messman on a Liberty Ship that was torpedoed, and he, Cisco Houston, and Jimmy Longhi were re-patriated to the states through the UK. The whole story is told in Longhis' wonderful account WOODY, CISCO, AND ME."

Correct, and the episode was described in the Arena Woody Guthrie documentary. The liberty ship was actually a troop ship carrying personnel for the Normandy invasion. Further details are in Joe Kleins Woody Guthrie; A Life, and Ed Cray's Rambling Man. Both highly recommended.

The Childrens Hour presenter makes it sound as though Guthrie did the whole programme. While that's unlikely, BBC Radio 2 broadcast a four part series on Guthrie many years ago and included a 5 minute clip from that show. I can't remember whether Car, Car was part of the clip, but the introduction about washing dishes on a liberty ship certainly was. So too was Woody singing Wabash Cannonball.