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The Wand of the War Mage gives a bonus to spell attack rolls:

While holding this wand, you gain a bonus to spell attack rolls determined by the wand’s rarity. [+1, +2 or +3]

The summoning spells from Tasha's Cauldron of Everything summon an ally that uses the caster's bonus to spell attack rolls when attacking. For example, the Summon Fey spell summon a Fey Spirit with the following shortsword attack:

Shortsword. Melee Weapon Attack: your spell attack modifier to hit, reach 5 ft., one target. Hit: 1d6 + 3 + the spell’s level piercing damage + 1d6 force damage.

Does holding the Wand of the War Mage increase the accuracy of summons such as the Fey Spirit when attacking? If I later drop the wand, would the summoned creature suddenly find itself less accurate?

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Wand of the War Mage does not impact your summons

Your spell attack modifier is defined as follows (from the wizard class and the chapter on Spellcasting):

Spell attack modifier = your proficiency bonus + your Intelligence modifier

Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

Note, it does not say that situational pluses and minuses change your attack bonus. Thus, Wand of the War Mage does not impact your attack bonus, it simply gives you a bonus to your attacks.

That said, things that change your spellcasting ability modifier or your proficiency bonus would impact your attack bonus, but whether these sort of changes impact your summons is not clear. A GM can rule either that the "your spell attack modifier to hit" clause is checked only once, when casting the spell, or they can rule that it is checked constantly, and any changes to it impact the ability of your summons.


A GM can always rule otherwise

A GM is also well within their rights to say that Wand of the War Mage does impact your spell attack bonus, after all, it gives a bonus to your spell attacks. Just note that this means any other situational modifiers (for example, from the resurrection, raise dead, temple of the gods, bless, and synaptic strike spells) would apply as well.

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