The exclusion of any mention of magical weapons isn't accidental, Barbarians are resistant to bludgeoning, piercing, and slashing damage whether it's magical or otherwise as it's key to their ability to have staying power on the front line. By comparison, the Heavy Armor Mastery feat makes the clear distinction for magic weapons, therefore, it's not unreasonable that the design intent is on purpose.
This is important because otherwise the Barbarian would experience a huge drop off in survivability as the game progressed passed a certain tier when a lot of enemies' weapons are being described as inherently magical.
The quote you give for the effect of blade ward lacks the most important word in answering this question:
Until the end of your next turn, you have resistance against bludgeoning, piercing, and slashing damage dealt by weapon attacks.
"Weapon attacks" is a well defined phrase in D&D 5e, and refers to all things which are attacks and not spells, approximately. The important thing here is that some thing cannot be a weapon attack if it is not an attack; "weapon attack" is a special type of "attack".
Whether or not a trap makes an attack is covered by a previous question. Short version: if it makes an attack roll, then it's an attack. If the description states that it is an attack, using the exact word "attack", then it's an attack.
As mentioned earlier, some spells also require attacks, and those are "spell attacks". Other attacks are "weapon attacks". I haven't found a perfect explanation for how to distinguish 100% of the time, but I would say that if a spell calls for an attack roll, or an ability uses the exact phrase "spell attack", then it's a spell attack. Note that attacks with magic weapons are still weapon attacks even if they deal magical damage.
If a trap meets both requirements, i.e. it is an attack and it is not a spell attack, then blade ward would provide resistance against its bludgeoning, piercing, and slashing damage.
For example, here is the first trap I could find on dndbeyond, the Hunting Trap:
A creature that steps on the plate must succeed on a DC 13 Dexterity saving throw or take 1d4 piercing damage and stop moving.
The Hunting Trap requires a saving throw in order to avoid the effect. It does not make an attack roll, so it is not an attack and blade ward will not grant resistance against its damage.
Best Answer
I think they count as magical in three cases, but not in all others.
The berserker axe and hammer of thunderbolts (DMG), and another weapon from Hoard of the Dragon Queen, are always magical
For concreteness, let's look at the berserker axe, the first attunement-requiring weapon in the DMG. Its description reads, in part (DMG p. 155, emphasis mine):
Grammatically, it would seem that only the HP-benefit requires attunement, and that the +1 bonuses apply the moment you swing it. Then, by dint of having a magical +1 to attack and damage, I'd say the attack is magical and cuts through the shadow's resistance. (groan)
The hammer of thunderbolts also has this "severable" construction to its features, as does (HotDQ spoiler)
These two are definitely a close call, though. Perhaps even a toss-up.
In all other cases, the item only grants its nonmagical benefits
Most attunement-required weapons don't have this sort of provision, and it would seem that all of their magical utility requires attunement. See, for example, the holy avenger, flame tongue, staff of _____, sun blade, sword of _____, etc.
Further, the "Attunement" rules include the following statement (DMG, p. 138):
Certainly it looks like all of the other weapons, then, wouldn't give you their magical benefit without attunement. Moreover, the berserker axe does, in fact, require attunement, so this would seem to argue that you don't get any magical benefits from the axe. So perhaps even the berserker axe and hammer of thunderbolts need attunement to cut through resistance.