One D&D will bring about various rules revisions, some large, some small. The game still basically works the same way - we're not going back to THAC0 or replacing proficiency with skill ranks.
One of the most popular changes is also a rather subtle one.
In the current rules, nothing is stopping anyone with Extra Attacks from fighting with two different weapons - attacking with your Flame Tongue Longsword in one hand with attack number 1 and your Frost Brand Rapier in the other hand with attack 2. But light weapons have a special property, which allows you to make two attacks with your main hand and then get an additional, bonus attack with a weapon in the off hand - allowing those who don't have extra attack to make two attacks and those who do to make three (or, if you're a fighter, possibly 4 or 5, depending on how many extra attacks you get). The only downside is that you don't get to add your Strength (or Dexterity, with a finesse weapon) to the damage roll.
That's not actually changing.
Again, this is a subtle change. What is changing is that, under the current rules, that bonus attack uses your bonus action.
Depending on your class, this might be no problem, or it could be a bit of a hinderance.
However, under the new One D&D rules, the light weapon bonus attack is now just part of the attack action, leaving your bonus action free.
What's the real implication here?
What I want to go through is how this is likely to affect each class that likes to fight with weapons and to see how powerful this can be.
Artificers:
Two of the four Artificer subclasses favor weapons, but the Armorer is rather locked into using their built-in armor weapons, neither of which has the light property. However, the Battle Smith does need to use their bonus action more or less every turn to command their Steel Defender to attack.
Artificers don't get Fighting Styles, but an Artificer can pretty easily get magical weapons. Even if they have a stingy DM who doesn't give them any magic weapons, the Battle Smith could, by level 6, have one light weapon infused with Enhanced Weapon and another infused with Radiant Weapon, giving them at least a +1 bonus on the damage of each weapon.
Still, the ultimate bonus damage here might not be all that exciting. We can do better.
Barbarians:
Ok, now we're talking. The Barbarian, of course, typically starts an encounter by going into a Rage, which takes a bonus action. Barbarians love getting more attacks, because their damage is boosted by Rage in addition to their Strength and any magical bonus the weapon has.
Under the current rules, a dual-wielding Barbarian will suffer a bit of a drop in damage compared to someone using a two-handed weapon in short fights, due to the fact that they miss out on that first bonus attack. But this downside will disappear - partially - with the new rules.
For example, let's imagine a level 9 Barbarian with +5 to Strength, using a Greataxe versus dual-wielding Hand Axes. At this level, they have a +3 damage bonus from Rage.
Under the new rules, every turn, not just turns after Raging, will look like this:
2H: 1d12+8, or 14.5, times 2, which is 29
Dual-Wield: 1d6+8, or 11.5, times 2, so 23, plus 1d6+3, so 29.5.
Notably, until hitting level 9, this does not outpace the two-hander. (I'll also note that if Barbarians get Fighting Styles, as they look likely to, the total damage bonus of taking Two Weapon Fighting will be 5 per turn, while Great Weapon Fighting with a Greataxe will be about 2.67, which should let dual-wielding pull ahead. I believe that Brutal Critical will actually balance out as, while your critical with a short sword will not as impressive, you also have a 50% higher chance to get a critical with an extra attack.)
So, Barbarians, at least by mid levels, will win quite a bit here.
Fighters:
Fighters have always devalued dual-wielding because they already get so many attacks. If you only get two attacks per turn normally, an extra one is 50% more attacks. But if you're attacking 4 times a turn (granted, that's only level 20 Fighters - though we'll see about that given how everyone's getting Epic Boons at 20 now) that's only 25% more. One thing we also have yet to see is how this will interact with Action Surge - I believe the wording still says once per turn, which would seem to have Fighters further devalue light weapons.
Monks:
This all depends on whether a Monk's bonus action Martial Arts unarmed strike remains a bonus action or gets similarly folded into the attack. If one could do both of these, it would allow you to attack four times a turn at level 5, quite a bit more impressive than Fighters having to wait for level 20.
Currently, the Martial Arts attack functions kind of like built-in dual-wielding along with the fighting style. If it's a choice between a Shortsword with no damage bonus (a straight d6) versus even a tier 1 monk's unarmed strike (1d4 plus probably 3) the unarmed strike will do more damage.
We just need to see how things are designed here.
Paladins:
Paladins currently don't have access to two-weapon fighting as a fighting style. But even without it, a dual-wielding paladin build is not crazy, especially at high levels. First of all, dual-wielding lets you fish for critical hits, which you can then drop a Divine Smite on. At level 11, every weapon attack you make does an extra d8 of radiant damage, which durns a potentially dinky 1d6 light weapon into 1d6+1d8, or about 8 extra damage on average (Polearm Master is great on Paladins as well, as the bonus strike may only be a d4, but with the extra d8 and getting to add your strength modifier on top of any magic bonus).
That being said, Paladins don't really struggle for bonus actions in most cases - there are the specialized smite spells, and depending on how liberal you are with such spells, you might feel better about this, but I think you won't see a huge change here.
Ranger:
For opening rounds, this can potentially be a game changer, because Hunter's Mark is now being positioned as a central Ranger mechanic (not that it wasn't before). I can definitely say that my Gloomstalker dual-wielder would sometimes feel frustrated that things were dying each turn, and so he'd only get a turn on something with Hunter's Mark, and not be able to take advantage of doing 2d6 damage with his main hand and off hand.
The point is, Hunter's Mark is the sort of spell that gains value for every attack you make, but is also rather demanding of your bonus action, as you'll often need to use that to move the effect from one target to the next as your foes go down. Freeing up your bonus action is going to make this way less painful.
I'll also note that the Crossbow Expert feat has been updated with this in mind - sort of. The current version lets you make an attack with a hand crossbow (or now, a crossbow with the light property, which is at least currently just those) as a bonus action after you make an attack with a one-handed weapon. Here, the implication is that you can always wield a hand crossbow and get the same benefit of the light weapon, but this lets you add your ability modifier to the damage. This might actually be a nerf, as I don't know that this will allow you to use the same crossbow for the "normal" attacks and the bonus, light weapon attack. (Also, Jeremy Crawford has said that "ignoring the loading property" only allows you to use it with something like Extra Attack, and still expects you to have a hand free to reload). Indeed, this could either be a buff or a nerf, as on one hand it could imply that even if you have, say, a rapier in one hand, you don't need that free to load the hand crossbow. But it also seems to imply that if you have, say, a souped-up +3 Hand Crossbow, you need a second to fire off the off-hand attack.
Rogues:
Rogues might be the big winners here. Dual-wielding as a Rogue is generally a really good idea, because it significantly increases the chance to get sneak attack. The actual damage of the off-hand attack is kind of inconsequential, because you're getting way more out of it when you get a chance to get your sneak attack after missing on the first attack.
Let's imagine a scenario to demonstrate: We have a +5 to hit and are attacking something with an AC of 16, meaning we have to roll an 11 or higher to hit the target (and obviously critting on a 20). We'll say we're level 3, so our sneak attack is 2d6. We're using Daggers because we're being stylish rather than optimized.
So, our "hit array" is:
Miss 1-10 (50%), Hit 11-19 (45%), Crit 20 (5%)
We start by calculating the damage when just attacking once (and we're assuming we've got an ally in 5 ft of the target and don't have advantage).
So, our damage will be 1d4+3+2d6, or 5.5+7, aka 12.5. Crit damage is 2d4+3+4d6, or 22 on average.
So, 45% times 12.5 is 5.625 and 5% times 22 is 1.1, meaning our total average damage per attack (in a single-attack scenario) is 6.725.
But, when we dual-wield, we calculate this differently. Sneak Attack can trigger on either attack, but only once per turn. Thus, if we miss with the first attack, we can potentially get the damage on the second. The way we calculate this is the following: We start by simply calculating the weapon damage for the first attack on its own and subject it to the same Hit/Crit chance coefficients to find our average weapon damage per first attack. We do the same for the second attack (which will be less as we're not adding our Dexterity modifier to the off-hand damage).
However, for the Sneak Attack, we start by applying the same to its 2d6, which will account for the first attack. Because a hit on the first attack means no sneak attack on the second, we need to basically imagine a nested, microcosmic scenario where we miss the first attack and then apply the same thing.
Essentially: Our chance to get sneak attack on the second attack only happens if the first misses, and so we're only working in the 50% of the time that the first attack misses. Within that scenario, we then have the same chance as before, so we take our usual 45% hit chance times hit damage and 5% crit chance times crit damage and then multiply it by 50% to account that this is only ever happening if the first attack misses.
The ultimate upshot of this is that Sneak Attack in this case goes from missing 50% of the time to only 25% of the time, which should increase our potential damage significantly. Let's calculate it out:
Main Hand Weapon Hit: 1d4+3, or 5.5
Main Hand Weapon Crit: 2d4+3, or 8
Damage per first main weapon attack: 45% x 5.5 + 5% x 8, or 2.875.
Off Hand Weapon Hit: 1d4, or 2.5
Off Hand Weapon Crit: 2d4, or 5
Damage per off hand attack: 45% x 2.5 + 5% x 5, or 1.125 + 0.25, or 1.375
Sneak Attack Hit: 2d6, or 7
Sneak Attack Crit: 4d6, or 14
Main Hand Sneak Attack Damage Per Attack: 45% x 7 + 5% x 14, or 3.15 + .7, or 3.85
Off Hand Sneak Attack Damage Per Attack: 50% x that last number, or 1.925
Now, we add them all up and find that our total damage average per round is 10.025.
That's a huge jump from the 6.725 when attacking with just one weapon, and most of that difference is accounted for in the bump to sneak attack.
Now, what does this matter? Rogues can already dual-wield light weapons. The key here I think is that Rogues are also a class that really likes to use bonus actions - there's basically no reason not to use a bonus action on your turn thanks to Cunning Action at 2nd level.
So, now you can get these two attacks in and then potentially hide to get advantage on a subsequent attack or dash or disengage to position yourself more favorably. One popular move is to run up to a foe that is within melee of an ally, attack, and then disengage, forcing them to chase after you (and draw an opportunity attack from your ally) if they want to punish you.
So, in conclusion, I'd say that our probable biggest beneficiary of this change is the Rogue, though I think that Rangers and Barbarians are going to be very happy about it as well. (I'm curious to see if there's any errata for subclasses printed outside of the PHB - the Soul Knife would probably love to be able to get their extra psychic blade as part of the attack. That does have the Dex bonus built into it, though, so who knows.