I've seen lots of people do this, and unless it was clearly stealing en-masse for profit I doubt there'd be a problem, but is it legal to pick up dead logs, kindling etc. and take them away to use elsewhere as firewood?
All land in the UK is owned by somebody, therefore, all trees and their produce (including firewood) are owned by somebody. You could be charged with theft if you take logs, kindling etc. without permission.
The Jeremy Vine Show on BBC Radio 2 on 6th January 2012 discussed this very topic following the storms in the UK around that time, that left a lot of fallen trees. Jeremy Vine interviewed a lawyer who I remember explained the laws that cover this issue.
The Forestry Commission sometimes offers Firewood Scavenging Permits allowing you to gather your own firewood, usually from an area that has already been commercially harvested.