I did some light trail riding with a Shimano Nexus 3sp c/b and it held up fine. You do want to find a frame with horizontal dropouts in order to avoid having to use a chain tensioner.

I don't know how well the Sturmeys are sealed. It is possible grit would get in there somehow especially if it wouldn't get stuck in grease before doing any harm. It would circulate through the oil on an old Sturmey.

They offered a heavy duty, offroad AW during the early bmx craze in the 80s. I think they would really prefer you didn't put too much hard offroading on the standard ones. Shimano says so too in their tech pages on their site.