How was Churchill able to drink so much so often and yet still retain popularity and credibility as a public figure?

I think there are two reasons. First, alcohol consumption was not nearly so disrespected then as it is now, especially in Britain. If a man could hold his liquor, that is it did not interfere with his job, then it was entirely acceptable for him to drink. Indeed, people who didn't drink, like Montgomery, were considered rather odd. And the second was that, however much he drank, it never interfered with his job. Whether his metabolism gave him extra tolerance, or his high consumption allowed him to develop high tolerance, he was never "drunk on the job" so to speak.

There is, perhaps, at least a third reason. Hitler was known to be a tea-totaller. Soldiers are known for their propensity to consume alcohol. Thus, allied troops identified with Churchill and his drinking, while having distain for Hitler and his abstinence.