You can get some statistics after the problem is solved.

nintvarsNumber of integer variables
nbinvarsNumber of binary variables
nnodesNumber of explored nodes in branch and bound
ncutsNumber of cuts
nbranchesNumber of branches
nlevelsDeepest level reached (Root node is level 1)

To access these statistics you can use JuMP.backend(m) i.e.:

internal_model = JuMP.backend(m)
println("#IntVars: ", internal_model.optimizer.model.inner.nintvars)