Almost every wand that casts a spell (eg Wand of Fireballs, Wand of Lightning Bolts, and Wand of Magic Detection) contains the wording:
While holding it, you can use an action to expend 1 [or more] of its charges to cast the [spell] […] from it.
For these wands in particular, can you use the wand to cast a spell after having cast a bonus action spell earlier on your turn?
The rule on bonus action spellcasting makes it clear that you can only cast a cantrip with a casting time of 1 action after having cast a bonus action spell.
However, does casting a spell from a wand still fall under this restriction? Does it still count as "casting a spell" for that purpose? Is this a case where the specifics of the magic item override the general rule?
Note that this question is based off of, but not a duplicate of Can you use a Wand of Wonder after casting a spell as a bonus action? because the Wand of Wonder does not have the same wording and arguably makes it a different case.
Not if the spell being cast is 1st level or higher.
Per the spellcasting rules (p202 PHB), if a spell is cast using a bonus action, only a cantrip may be cast using your action.
This section makes no reference to expending spell slots, only the casting of spells. The wands should still be covered by this general rule.
But are YOU casting a spell when you use the wand, or is the wand? The Wand of Detect Magic states
Here, a charge is analogous to a spell slot. You use a finite resource and as a result you cast a spell. The "from it" part could sound like the wand is doing the casting, however this is probably to signify that the wand itself is being used a focus for the spellcasting. The spellcaster is still the one casting spells, not their focus.
Under the Magic Item section of the DMG, on p141 there is a section regarding items that when activated casts spells.
This further leads credence to the point that the user is casting the spell as opposed to the item.
Also on p141 of the DMG, under Activating an Item.
Activating one of the wands doesn't come under the Use an Item action.
It appears that the general rule for bonus action spell casting still holds true for using magic wands. As the user is the one casting the spell, they are limited to the use of wands that produce cantrip spells, if they have already cast a spell with their bonus action.
As an aside, allowing the use of wands in this manner would definitely skew the action economy in favour of the spellcaster. Wands are typically an inflexible but rechargeable source of extra spell slots, which can be quite powerful. They free up prepared spell lists and set aside additional slots for use. Allowing them to circumvent the standard spellcasting restrictions as well would make them even more powerful. The first cheese tactic I can think of is the spellcaster getting surrounded in melee combat, then Misty Steping away and Fire Balling the crowd of mobs before they have a chance to react.