As said by others, the problem doesn't arise if you think it's a bubble head scenario or a thin suit of air. The problem seem to arise when you think of the air around you as something more like the shape/size of the "core" aura of Dragon Ball characters.
So let's go through things:
Bubble head: Good solution, but unrealistic: the necklace protects the wearer from toxic fumes, so a bubble head would not protect you from acid fumes that would burn the rest of your skin. Unrealistic interpretation, discarded.
Thin suit of air: If this is the case, then you'd be able to swim normally.
DB-like aura: now things get interesting. There are two options about HOW the necklace works, and they affect the result dramatically:
Option 1: The necklace magically transmutes the medium around the wearer into clean fresh air: This is bad. You are basically creating a space devoid of water that is now air, the density changes a lot, you fall down (and, since your maximal speed in air is greater than in water, you'd slam on the ocean floor and take a lot of dmg if you jump out of a boat).
Option 2: The necklace basically makes you "wider": only fresh air and no other medium can exist in 30cm around you. This is actually interesting: you have just gained a LOT of water floating power! As per Archimede's principle, you are now moving a lot more water, which is heavier than air, so the upwards resulting force is stronger than without the necklace! Since you can float without it, you can float even better with it! Though normal swim movements would likely be impaired by the unusual situation, but compensated by your increased surface (your increased surface acts like flippers). In this case I'd rule out that you swim at your normal swimming speed, but are twice as hard to sink.
Bottom line: it depends on how you define the necklace's working! Or, even better, take note of these ideas, then let the player describe the act and punish/reward him accordingly :)
Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered. Per the rules on climbing, swimming, and crawling:
Each foot of movement costs 1 extra foot (2 extra feet in difficult
terrain) when you’re climbing, swimming, or crawling. You ignore this
extra cost if you have a climbing speed and use it to climb or a
swimming speed and use it to swim.
So movement is ⅓ if the heroes are in difficult terrain - normal swimming is only ½.
Next, spellcasting. There are no specific rules about underwater spellcasting, but the description of verbal components says:
Most spells require the chanting of mystic words. The words themselves
aren't the source of the spell's power; rather, the particular
combination of sounds, with specific pitch and resonance, sets the
threads of magic in motion. Thus, a character who is gagged or in an
area of silence, such as one created by the silence spell, can't
cast a spell with a verbal component.
It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.
Finally, breathing mechanics. The rules for suffocating are on page 183 of the PHB, and are much simpler:
A creature can hold its breath for a number of minutes equal to 1 +
its Constitution modifier (minimum of 30 seconds).
When a creature runs out of breath or is choking, it can survive for a
number of rounds equal to its Constitution modifier (minimum of 1
round). At the start of its next turn, it drops to 0 hit points and is
dying, and it can't regain hit points or be stabilized until it can
breathe again.
For example, a creature with a Constitution of 14 can hold its breath
for 3 minutes. If it starts suffocating, it has 2 rounds to reach air
before it drops to 0 hit points.
Best Answer
So, firstly, water has no effect on sovereign glue. This isn't a bottle of clag paste; this is a legendary magic item. DMG, page 200:
Unless the well is full of universal solvent, it won't affect the glue in any way.
Now, that said, this trap probably won't work. The glue takes a minute to set, and only binds unbreakably after that time. So unless the glue is placed on the sword just before someone tries to grab it, and they're still holding it when the glue finishes setting, having held it for a full minute 20 feet underwater, the glue won't bind to their hand. Instead, the sword's hilt will be unbreakably bonded to... nothing.