The bonus does apply.
Firstly, as stated in Purple Monkey's answer, the item doesn't specify the origin of the spell attack roll as a qualification for the bonus. So, the bonus then applies to all spell attack rolls, regardless of the origin. The Wand also doesn't say that you need to take the "cast a spell" action to gain the benefit. So, we don't need to worry about the "use a magic item" scenario
But what if the spirit of the wand is that it only works on spells cast by the wielder?
Even in this case, the bonus still applies, because the language on all spells being cast from items is "you cast..." meaning that even when casting with an item, it is ultimately still the wielder that is casting the spell. This is the exact reasoning used in this answer to a similar question about using class abilities for spells cast through items.
Even sage advice suggests this is the proper stance to take
There are two relevant sage advice questions that help draw this conclusion. One about drawbacks from casting Wish from an item, and one about concentration on a spell cast from an item.
In regards to the question "Does casting Wish from a magic item (scroll, Luck Blade, Ring) have the same negative effects as the spell?" The answer was:
yes, if the item specifies that you are casting the spell and makes no other exceptions. 1
In regards to the question "If a magic item casts a concentration spell (e.g. bless - Necklace of Prayer Beads), does the PC have to concentrate on it?" the answer was:
Yes, unless the item says otherwise.
So, it's clear that unless the item description specifically states that the item rather than "you" are casting the spell, the spell is being cast by the wielder, not by the item. The item is supplying the enery for the spells, but it is the caster who is actually manipulating that energy into effect. Thus, the wielder should also be able to draw on the power of the Wand of the War Mage to get the relevant bonus since they are the source of the spell.
This answer was supplied by Mike Mearls, who is typically not given as much authority as Jeremy Crawford. However, Crawford chimed in on that sage advice for extra clarification and did not correct Mike in any way, effectively agreeing with Mike's ruling.