How can I get the number of branch and cut nodes explored within the script?

Use the solve details info contained in Model.solve_details. More precisely:

  nb_nodes = mdl.solve_details.nb_nodes_processed

See http://ibmdecisionoptimization.github.io/docplex-doc/mp/docplex.mp.sdetails.html#docplex.mp.sdetails.SolveDetails.nb_nodes_processed