Conclusions from reverse mathematical strength of graph minor theorem
I am not sure I understood your question, but if you are asking how difficult it is to compute the set of obstructions, you may be interested in the following http://www.jucs.org/doi?doi=10.3217/jucs-003-11-1194 where the non computabubility is proved even if the graph class is MSOL-definable. In this paper http://www.sciencedirect.com/science/article/pii/S0012365X97830798?via%3Dihub the computability is proved when the graph class is given by the HR grammar.