If you're asking if culture at the southern end of your empire will help push your northern borders, then no, it won't. Culture expands radially from each of your cities- the furthest a tile can be from a city and still be affected by its culture is 6 tiles (and only if that city has Legendary culture).
However, if you have two or more cities producing culture on a tile, I think both of them will "work together" for your empire in order to claim a tile from a rival. Examle: in the screenshot you posted, let's say that some AI dropped a city on the ice 1 tile NE of that mountain, denying you that Fur that's just outside Oxford's and Newcastle's BFCs. When either one of those cities pop their borders, you'd start getting culture in that tile. When both cities pop their borders, you'd get Newcastle's culture plus Oxford's culture against the one city that the AI has- making it easier to take that tile. So basically whatever you have going on on your northern border, if you can get two (or three) cities all producing culture in a tile, it makes it that much easier to take it.
(Experts and code-divers, if I'm wrong on that second part, please let me know...)