You cannot use the wand more than once per turn with these two features
Using a wand is a special action defined by the wand's description. The prototype wand says:
As an action, you can cause the magic wand to produce the cantrip [...]
You cannot produce a cantrip from a wand with the Attack action.
When you take the Attack action, Arcane Armament allows you to attack two times in that one action. Attack meaning to attempt to hit something with a weapon. But you cannot replace an attack with another action. And using the wand to produce a cantrip is not considered, in and of itself, to be an attack. Cantrips may have attacks as part of their effects, but producing a cantrip is not itself an attack.
When you produce a cantrip from the wand you have to use your full action to do so, which means normally giving up the option to Attack.
So, you cannot take an attack given to you by the Attack action and use it to use the wand to produce a cantrip.
These features should be compatible, though there is some awkward wording
The first important thing to note is that the Enhanced Wand infusion doesn't require you to actually use the wand to do anything, it requires only that you are holding the wand when you cast a spell:
While holding this wand, a creature gains a +1 bonus to spell attack rolls.
It's basically a prototype version of the wand of the war mage which has the same wording:
While holding this wand, you gain a bonus to spell attack rolls determined by the wand’s rarity.
You don't need to use the wand as a focus or actually involve it in the spell in any way. Though one imagines that is the narrative intent, the rules only require that it's in your hand when you cast a spell. Wands generally being one-handed tools, you are free to hold something else in your other hand, such as another wand, and use that normally too.
You will need to keep in mind that you need a free hand to manipulate material components and/or provide somatic components for spells, though; specific wands might or might not count as an arcane focus depending on your DM's ruling, which could obviate the problem for spells with inexpensive and unconsumed components, but for spells with significant material components - or spells with somatic but not material components - you can't get around needing a free hand.
The Artillerist's Wand Prototype feature basically lets you create a magic wand that only you can use which you can use to cast a cantrip. As explored in this question, casting spells from magic items is in most respects the same as having cast the spell yourself - you get the benefits of class features and abilities that interact with spells you cast, and you should also get the benefits of magic items like the wand of the war mage.
The only issue I can see is that the Wand Prototype description doesn't actually state that you cast the spell using the wand, it says that:
As an action, you can cause the magic wand to produce the cantrip, using your spellcasting ability modifier
But, keeping in mind that the Artificer is still UA material, 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; it should still count as you casting the spell for such purposes, and get the benefit of a wand of the war mage (or an Enhanced Wand made by infusion) if you happen to be holding one in your other hand.
Best Answer
An Artillerist adds their Intelligence modifier as a bonus to the total damage rolled per hit.
The relevant text in the Artillerist subclass says:
The bonus is added when the Artillerist rolls any amount of dice to calculate damage. The bonus is added once to the total. "Damage rolls" are defined in the combat rules:
With Fire bolt, and most other cantrips, the Artillerist would make one attack roll. If it hits, they make one damage roll, and add their Intelligence modifier. At 5th level, that roll would be 2d10 + (Intelligence modifier) fire damage.
Although Eldritch blast involves multiple damage rolls, it's not a valid candidate because the Wand Prototype feature only applies to artificer cantrips:
However, if you had some way of making Eldritch blast an artificer cantrip, then the Artillerist would add their Intelligence modifier to each attack's damage roll.