What you're referring to is in fact an "A99 aquata breather". In-canon these are described as a "Jedi Breathing Device".
The official Star Wars Databank has an article about them:
A small breather that fits snugly into a Jedi Knight's utility belt, the A99 aquata breather is a compact mouthpiece designed to supply its wearer with a breathable gas. The A99 has advanced filters to allow it to function underwater, in a vacuum, and in certain poisonous environments. The A99 has a number of variants to accommodate a variety of alien physiologies. The A99's miniature compressed air tanks can supply two hours of oxygen.
As you can see from the images below, Obi-Wan is still definitely wearing his lightsaber.
And here's a closer look at the device:
And the actual prop used on screen by Qui-Gon (from the Lucasfilm Archive exhibition)
And from the Phantom Menace novelisation:
The Gungan pointed at the lake. “We go underwater, okeday?”
The Jedi looked at each other, then extracted small containers from their clothing, releasing portable breathing devices the size of the palms of their hands.
“Me warning yous.” Jar Jar’s eyes shifted from one to the other. “Gungans no like yous outlanders. Yous not gonna get warm welcome.”
and the Revenge of the Sith novelisation, which specifically confirms that lightsaber belts do indeed work underwater.
...he was vaguely pleased to discover that even in his semiconscious fall, he’d managed to hang on to his lightsaber.
He clipped it back to his belt by feel, and-using only a minor exercise of Jedi discipline to suppress convulsive coughing-he contracted his diaphragm, forcing as much water from his lungs as he could. He took from his equipment belt his rebreather, and a small compressed-air canister intended for use in an emergency, when the breathable environment was not adequate to sustain his life.
Obi-Wan was fairly certain that his current situation qualified as an emergency.
Interestingly, the original concept was that instead of a breather device, the Jedi were going to put jellyfish on their faces(!);
U.K. Concept Artist Kun Chang helped develop the look of the aquata breathers. "One idea for the breathers that everyone loved -- but ultimately wasn't used -- was that Obi-Wan and Qui-Gon were going to pick up a kind of jellyfish and place it over their faces which would enable them to breathe underwater," reveals Kun.
