Though I'm not Catholic so I know little of this history, it is interesting.
There's a *great* Hughes as President TL that was posted on soc.history.what-if a number of years ago. This one would probably be a bit different, though. That one has a close election with Wilson giving his position up to Hughes a touch early becasue it is certain the U.S. will be going to war (I forget who wrote it, but those familiar surely recall it), but I don't know if Hughes would co-ioperate if Wilson has lost votes this way. There's not the guarantee that the U.S. will enter the war.
I think one thing you might get is more Democrats favoring Al Smith. He was close to the nomination in 1920, but lost out in favor of James Cox, and didn't win till 1928.
Here, Democrats would realize that Wilson's anti-Catholic sentiments cost them the White House. Wanting it back, they would try to support Smith more. I don't know if this means he'd easily be the nominee in 1920, that would depend on how positive the Pope's job is viewed in Catholic eyes, and also whether he's still alive.
If Gibbons is still alive, no way, because they'd need to satisfy the Proestants who would worry about America becoming too closely tied to the Vatican. (Remember, this was a worry in 1960 as it was in the general election, and in 1920 probably denies Smith the nomination.) If Gibbons died in the interim and was seen as doing a very good job, though, Smith might be the nominee, as they feel they can try to win Catholic voters with Smith, and that non-Catholics will not fear a connection to the Vatican too strongly - while at the same time feeling both will see an American Catholic can do a great job on the world stage. (This is not, of course, to say that the general populace won't fear it, only to say that those doing the nominating will feel it's safe.)
If Gibbons dies in the intervening 4 years, and only is seen as having done an okay job, Smith might be Cox's VP nominee to get Catholics back to the party. Which might make things interesting for FDR, who would not have been in the national spotlight, whereas in OTL Roosevelt was Cox's VP choice in 1920. OTOH, he wasn't a first ballot choice in OTL 1932, so that might stay the same.
Either way, I would imagine Hughes still wins re-election.