While pointing at someone
That's him swimming.
or
That' he swimming.
While pointing at someone
That's him swimming.
or
That' he swimming.
"That's him swimming" is the only form I've heard in my life in any of the varieties of English I've heard.
I've never heard "that's he swimming" in any situation.
"That is him swimming." can be thought of as an abbreviated form of:
"That is whom you can observe swimming." where 'whom' is the object of 'observe'.
The participle 'swimming' cannot take a subject, e.g.
I see he swimming. (incorrect)
I see him swimming. (correct)
According to Geoffrey Pullum, co-author of The Cambridge Grammar of the English Language:
The forms with nominative pronouns sound ridiculously stuffy today. In present-day English, the copular verb takes accusative pronoun complements and so does “than.” My advice would be this: If someone knocks at your door, and you say “Who’s there?” and what you hear in response is “It is I,” don’t let them in. It’s no one you want to know. [Link to Source]
As this implies, there do remain people who use the predicate nominative in English, since Professor Pullum’s inverse snobbery is fortunately not lethal.