Finding max flow in a undirected graph
What you describe can happen in a feasible flow. However the net flow across the (undirected) edge $\{u,v\}$ would be $0$. Also, you say that the original graph has flow $2c$, this is false since the amount of flow from a vertex $s$ to a vertex $t$ in a graph is not the sum of the flows across all the edges, but rather the amount of flow leaving $s$ or, equivalently, entering in $t$.