As written, they do not combine/stack
You can definitely wield two wands, but because of the way the Wand Prototype is written, the +1 effect of an enhanced wand (or wand of the war mage) would not apply.
It's possible that because it's UA they didn't vet the language enough and ended up with a weird wording, but it's also possible that they purposefully made it restrictive. As written, it is straightfoward to me that you do not get the +1 benefit because you are not casting the spell; the wand is.
Some magic items allow the user to cast the spell from the item. The
spell is cast at the lowest possible level, doesn't expend any of the
user's spell slots, and requires no components unless the item's
description says otherwise. (DMG, p. 141)
For the same reason, you do not gain the advantages of the Spell Sniper or Elemental Adept feats when using the Wand Prototype feature.
Jeremy Crawford tweeted an answer to a similar question with regard to casting a spell through an item:
If a magic item's description says you cast a spell from it, you can use Metamagic on the spell.
It is my understanding that this tweet is addressing items which say "you cast."
All of the DMG wands I was able to quickly search have phrasing that is some variation of: "you can use an action to expend 1 or more of its Charges to cast..."
The wording of Wand Prototype reads: "As an action, you can cause the magic wand to produce the cantrip, using your spellcasting ability modifier."
If Wand Prototype were meant to function like all other wands, it should read: "you can use an action to cast the cantrip invested in this wand with this feature" or something along those lines.
Currently, the Wand Prototype feature reads (emphasis mine):
Whenever you finish a long rest and your woodcarver’s tools are with you, you can touch a nonmagical, wooden wand and turn it into a magic item. When you do so, you invest it with one artificer cantrip of your choice—even one you don’t know—that has a casting time of 1 action. As an action, you can cause the magic wand to produce the cantrip, using your spellcasting ability modifier (other creatures are unable to use the wand’s magic). The wand loses this magic when you finish your next long rest.
Any damage roll you make for a cantrip in the wand gains a bonus equal to your Intelligence modifier (minimum of +1).
Specific to Carcer's comment:
I think this is an unintentionally awkward wording and using the wand prototype is not supposed to be different than casting a spell from a normal wand in such a manner;
I'm unaware of any DMG or book wand that utilizes a spell attack. The rest call for saving throws or are magic missile. I don't think there's precedent to say that it should be assumed that the UA writers just made a mistake.