My instructor put in the notes a picture of a loop without an inductor - just a battery with voltage $\epsilon$, a switch, and resistor with resistance $R$. He explains that as soon as the switch is flipped, an emf is created that will oppose the change in flux. With an inductor, I can see how this would happen. But this made me wonder what happens when there isn't an inductor. Does current immediately reach $\epsilon\ /\ R$? Or is there some back emf that opposes flux change? It would make sense thinking about it that even without an inductor, there would be a resistance to flux change inside the closed loop, so it wouldn't immediately reach $I = \epsilon\ /\ R$.
The answer to your question is that there is always inductive coupling between the legs of such a loop. Beyond that, transmission line theory answers your question of what happens. Basically, there's parasitic inductance and capacitance everywhere you go, and they ensure you never see instantaneous jumps in current.