Two measures of effect size between an ordinal variable and a nominal variable are epsilon-squared and Freeman's theta. epsilon-squared is fairly-well described on the internet. Or see the function epsilonSquared. Freeman's theta is mentioned around the internet, but I think for the calculations, you have to get the Freeman 1965 book or see the function freemanTheta.
For a test of association, you might look at Cochran-Armitage test.