Kad's answer is correct -- any ball that has not touched the ground and is caught, regardless of the runner's position, is out by flyout. However, I wanted to challenge the premise of the question.
This is an impossible scenario
This could never happen in any baseball game due to the physics of a flyball. For example, this question has an answer for one ...
On September 18th, 2006, J.D. Drew, Russell Martin, and Marlon Anderson hit home runs on 3 consecutive pitches. Drew hit a home run off Jon Adkins. Martin and Anderson hit home runs off Trevor Hoffman.
I checked all of the instances of teams hitting 4 consecutive home runs, and this was the only one I found in which 3 were hit on consecutive pitches. ...
I have play by play data going back only to 1952, so this also an incomplete answer, but there have been 30 such incidences of back-to-back home runs, one tying the game, and one winning the game for teams in that time frame.
Here's a sampling from this decade:
July 26, 2017 - Blue Jays, Smoak/Morales
August 26, 2016 - Astros, Correa/Gattis
August 19, 2016 ...
As of 16 August 2017, the record stands at 8 consecutive games with a home run, and that is held by 3 people.
Dale Long - Pittsburgh Pirates - May 19 - 28th 1956 (8 Home Runs hit)
Don Mattingly - New York Yankees - July 8th - 18th 1987 (10 Home Runs hit)
Ken Griffey Jr - Seattle Mariners - July 20th - 28th 1993 (8 Home Runs hit)
He would be out. A home run is defined by a batter circling the bases in one play without being put out. Since the ball would still be in play even if he circled the bases, catching it would be an out.
Regarding over-the-fence bunt home runs, the minimum observed off-the-bat speed for a home run in 2016 is about 90 mph. The coefficient of restitution (COR) limit for MLB baseballs is 0.578. Assuming that a ball being bunted acts like the MLB's COR test, that implies a pitch speed of 156 mph (90 mph / 0.578) is required for a bunt home run.