How fast can we solve a totally unimodular integer linear program?
I believe On a class of totally unimodular matrices, by Yannakakis, gives an answer to your question for a special case of TU ILP (whenever there are no odd cycles in a bipartite graph obtained by seeing the coefficient matrix as an adjacency matrix). 

In that paper there is a reference to Polynomial algorithms for a class of linear programs, which seems to handle all totally unimodular matrices, but I am not sure about how much more efficient it is compared to generic algorithms for LPs.