I was reading up on the classes in D&D 5e. I saw that the 2019 UA artificer had a feature that allowed them to attack twice that wasn't the Extra Attack feature, called Arcane Armament.

The Arcane Armament feature allows a second attack during the Attack action so long as the second attack is made with a magic weapon. The pact weapon from the Pact of the Blade warlock meets this requirement.

Pact of the Blade warlocks can also take the Thirsting Blade eldritch invocation for yet another attack. I have read that using Extra Attack and Thirsting Blade doesn't result in 3 attacks.

Could I make 3 attacks in one Attack action, including one from Arcane Armament and one from Thirsting Blade? Or would it run into the same restriction as getting the Extra Attack feature from different classes?

With only a pact weapon, you can only make two attacks

Thirsting Blade says:

You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.

And Arcane Armament says:

Starting at 5th level, you can attack twice, rather than once,
whenever you take the Attack action on your turn, but one of the
attacks must be made with a magic weapon, the magic of which you use
to propel the attack.

Note that Thirsting Blade does not say that you get to attack "one extra time" with your pact weapon. It says you get to attack twice when you take the Attack action with that weapon. Thus, if you are using only that weapon to make your attacks, you will only be able to attack with it twice per Attack action. Arcane Armament just adds a different way to achieve two attacks - but neither ability will give you a third attack in this situation.

What if you make one attack with a weapon that is not your pact weapon?

Take this example: you made one attack with a non-pact weapon, dropped it then made the second attack of arcane armament with the pact weapon.¹

In this case, it actually creates a slightly ambiguous situation if you read it strictly by RAW.

By RAW, Thirsting Blade allows you to attack twice with your pact weapon specifically, and Arcane Armament allows you to attack twice in general; both effects trigger off of the above situation. If an ability is triggered, you should be able to benefit from its effects unless they are prevented by something else. The only way for both of those abilities' effect descriptions to be accurate would technically be to allow another attack with the pact weapon. And there is seemingly no rule preventing this from happening.

However, this is almost certainly not the intent at all. The fact that Thirsting Blade explicitly was added as an exception to Extra Attack (which is almost identical to Arcane Armament) tells us that the kind of interaction is not intended. Note that the artificer is in UA, and not all UA is balanced or tweaked for multiclassing. This is likely just a case of a loophole they haven't gotten around to addressing or noticing yet, but I'd expect it to be fixed upon official release.

^{1 - Two-weapon fighting would not work here, however, since the bonus-action attack is taken after the Sttack action and not as part of it.}

No, they don't stack. The rules specify that Extra Attack and Thirsting Blade don't stack, and that Extra Attack doesn't stack with itself, but it's actually not necessary to specify that; the wording of the features already prevents that from happening.

Extra Attack:

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

Thirsting Blade:

You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.

Arcane Armament:

Starting at 5th level, you can attack twice, rather than once, whenever you take the Attack action on your turn, [...]

Each of these lets you attack twice in an Attack action, instead of once. That is, all three features specify the number of attacks you can make in an Attack action; they specify "twice", not "one additional time".

If you have more than one of these features, you have multiple ways to attack twice in an Attack action instead of once, but you still don't have the ability to attack three times.