Returning to the topic - teaching fellows are available in many disciplines, but yes, without an EU citizenship, good luck with getting a visa. You're too low on the totem pole for most universities to bother to sponsor you, now that the government is limiting the number of Tier 2 visas. For a proper lecturer position, this won't be a problem, but it would be for a postdoc or teaching fellow.
At least at my uni, it still wouldn't be a problem for a postdoc. For any position that involves research, we appoint the best, independent of where they come from. No ifs, ands, buts, and no trouble from HR. For a teaching fellow, I'm not sure. Usually we only hire teaching fellows at short notice anyway, so our concern is more that we're desperate to get someone who can teach right away, and the visa thing, move, etc. could get in the way.