I think this question really is about how to find the p value given a U value for large sample sizes, since tables with critical values often stop at n1 = n2 = 20. The definition of the p value remains the same of course. For large samples, the U statistic is approximately normally distributed under the null hypothesis, with mean mu_U and standard deviation sigma_U. So the z statistic z = (U - mu_U) / sigma_U is approximately standard normally distributed under the null hypothesis. Hence, in order to find the (approximate) p value, you can compute the z statistic z = (U - mu_U) / sigma_U, and use the table for standard normal probabilities to find the p value given the z value. See

http://statkat.com/stattest.php?t=14 for how to compute mu_U and sigma_U, and for how to use the table with standard normal probabilities.