Can anyone advise if there is any day ticket water on (a) the Wear between Wolsingham and Witton and (b) the Tees between Middleton and Barnard Castle?

Hmm, I think a) some of that would be Bishop Auckland water and their (sort of) website seems to imply that they do day tickets. (There are contact details on the site so give them a ring.)

b) I believe the Raby Estate does day tickets for the Tees at Middleton and upstream from there, but I don't know whether their stretch goes down as far as Barnard Castle. There is however an excellent game/tackle shop in Barnard Castle who would most certainly know. Unfortunately, I haven't got a clue what they're called so unless someone else can step in you're going to have to google than one! (I think there's also a small free stretch at Barney too so you could enquire about that as well.) If you can't find them, then the Raby Estate has an office in Middleton which I'm sure would be easily googleable.

There's freefishing between the bridges.