Wand of the War Mage does not affect these spells.

"Spell attacks" and "weapon attacks" are mutually exclusive attack types (All attacks are one or the other, not both; See also), and the attacks made as part of these spells are melee weapon attacks; they use your normal attack stats/bonuses rather than your spell attack stats/bonuses, etc.

As such, effects that benefit spell attacks have no impact on the attacks for these spells.

Probably Not; but it depends on a finicky interpretation of the plain English of the feat

Let's look at the wording of the War Caster feat:

• You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.

—War Caster, Player's Handbook, pg. 170

I've highlighted the phrase "even when" because there are, by my interpretation, two valid ways to interpret this sentence in English, and which one you take seriously depends on how rigorously you interpret those two words.

In the former case, where those two words are superfluous, then no, you would be unable to cast Shield. In the latter case, where those two words are deliberate and important, then yes, you would be able to cast Shield.

"Weapons and Shields are now part of the subset of items that may be wielded when performing Somatic Components"

This is the interpretation that I suspect (but cannot prove) is intended by this feat, assuming the words "even when" are superfluous. The War Caster feat, expecting that spellcasters who take it will be in wartime situations, is afforded to allow spellcasters to be spellcasters while also being ready to pick up a weapon and defend themselves. So they specifically learn how to perform the intricate gestures associated with their spells while still holding their weapon/shield.

As a result, under this interpretation, you would not be able to perform the Shield spell while holding these items. Spellcasters must have a free hand to cast a spell that has no Material Components, even if the items in their hands would normally be valid as Material Components (like a wand serving as a focus). The War Caster feat would extend this to allow a Weapon or Shield to be valid items to be held while performing these gestures, but this would not be extended to anything else.

"You may now always perform the Somatic Components; Weapons and Shields are enumerated because they are notable"

If the phrase "even when" is semiotically important, it's because Weapons and Shields are de-facto presumed to negate a spellcaster's ability to perform their Spellcasting components—hence features like Improved Pact Blade for Warlocks, allowing their Pact Blade to function as a Spellcasting Focus. Therefore, when the text calls out Weapons and Shields, it's not to explicitly add them to a list, but to make it clear to a reader that they are intended to be included in the list even though a reader might implicitly assume they might not.

If this is the intended interpretation, then a better-worded version of this feature might have looked like this:

• You may now always perform the somatic components of spells; even when you are wielding a Sword or Shield in one or both hands

As a result, under this interpretation, you would be able to cast Shield regardless of what was in your hands.

The Former case is much more likely

Unfortunately, it's unlikely that the latter case was intended.

For starters, 5e has a very explicit "Specific beats General" rule, meant to be applied to any scenario where a specific feature might contradict the normal rules.

This book contains rules, especially in parts 2 and 3, that govern how the game plays. That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins.

Exceptions to the rules are often minor. For instance, many adventurers don't have proficiency with longbows, but every wood elf does because of a racial trait. That trait creates a minor exception in the game. Other examples of rule-breaking are more conspicuous. For instance, an adventurer can't normally pass through walls, but some spells make that possible. Magic accounts for most of the major exceptions to the rules.

—Specific Beats General, Player's Handbook, pg. 7

If the War Caster feat were intended to allow a spellcaster to perform the Somatic Components of their spells with any item, regardless of what it was, then it wouldn't be necessary to call out Weapons and Shields specifically: the following wording would have sufficed:

• You can always perform the somatic components of spells.

Secondly, the lack of punctuation in that original text implies a lack of emphasis on the predicate of the sentence as a distinct concept. The flow of the sentence is "You may perform the Somatic Components of a spell even when", not "You may perform the Somatic Components of a spell, even when".

"But Anything can be a weapon, right?"

Sure, pretty much any object can be treated as a 1d4+STR Improvised weapon. That doesn't necessarily mean the object in question is a Weapon, semiotically. Weapons are specifically defined as a category of item in the Player's Handbook, and the statblocks for individual items specifies whether it counts as a Weapon or not:

Martial Weapon, Melee Weapon

—Longsword, Player's Handbook, pg. 149

Whereas wands, excepting rare, specific scenarios, are not described as Weapons:

Spellcasting Focus

—Wand, Player's Handbook, pg. 151

Wand, Major, Rare (Requires Attunement)

—Wand of Fear, Dungeon Master's Guide, pg. 210

A non-weapon object used as an Improvised Weapon only counts as a Weapon while the attack is being made, and afterwards, mechanically, it is no longer a Weapon. So when a spell is being cast, it would not count as a Weapon.

Conclusion

So to conclude: Wands (and other objects that don't have Statblocks as weapons) don't qualify to be considered Weapons, even if you attempt to use them as Improvised Weapons.

So what this boils down to is the plain English interpretation of the sentence in War Caster; and by my reasoning, I don't believe the more permissive, more powerful interpretation is intended or correct.

So under the conditions of

• being a spellcaster
• with the War Caster feat
• that is holding two objects
  • one in each hand
  • neither of which is a Weapon or Shield

You would not be able to cast the spell Shield, which itself lacks Material components—hence why the Wand doesn't satisfy the "Somatic Components may be performed while holding a focus if the spell has Material components).