The Conjurer Wizard has a special place for me because my best friend in the first D&D game I ran (years before I ever got to play as a PC) made a High Elf School of Conjuration Wizard. It's a character I'd love him to bring back in a future campaign, and I think we'd both be happy with him updating to this new 5.5 version.
A lot of classes have what I'd consider the "quintessential" subclass - your Berserker Barbarians, your Devotion Paladins, your Open Fist Monks. Wizards, though... I don't know that they really have one. However, if you forced me to pick one, yeah, I might say Evoker, but Conjurers, even if they didn't show up in the new PHB, feel kind of like the classic Wizard archetype. After all, isn't it a fantasy trope for the king to have a "Court Conjurer?"
One of the funny things about Conjuration magic is that it focuses on two conceptually related but still very distinct things: one is teleportation magic, and the other is summoning magic.
Again, conceptually, I get that summoning things is like teleporting them (perhaps across planes,) so it does kind of fit together.
As far as I can tell, there weren't a ton of changes between the last UA with the revised arcane subclasses and the final version out of Arcana Unleashed, so I'm not really going to go in deep on every feature. I'm also... a little burned out on all my hypothetical "vs. Wyvern" calculations.
But I figure we can talk about this subclass a bit. I'm not going to give it a "tier ranking" but I did figure we could look at what you can do with it.
First off, Benign Transposition remains your main low-level feature. From 3-5, it is now basically a better Misty Step you can do Int times per day. While you can easily get Misty Step, especially with Conjuration Savant, there's actually a decent argument to not even grab it on level-ups, because you'll have access to this.
Here are the ways in which it's better to use this over Misty Step: First off, it's not a spell that costs a spell slot, so you can do this even if you took an action to cast a spell with a spell slot. Second, it can swap you with an ally. This is especially good if a monster broke off from a melee character to attack you - you can get away from the monster and then get your Barbarian friend in its face immediately.
Finally, at level 6, the range increases to 60 feet, making it twice the distance of Misty Step. Also, at 6 you can recharge its uses with 3rd or higher level spell slots. Now, Misty Step is only a 2nd level spell, so you've got to ask yourself if the extra distance, ability to swap with a friend, and the fact that it's not a spell (can't be countered, for example) is worth a higher level slot.
That being said, while Misty Step can be absolutely clutch when needed, I don't often cast it more than once or maybe twice a day. So I honestly don't think we will need to really worry about the spell slot cost to recharge.
But it's at 6 that the thematic focus of the subclass starts to shift over from the teleportation stuff to the summoning stuff.
Durable Summons is good. Quite good. Let's pick a Summon spell. We'll say Summon Fey, which is one of the better ones (Summon Undead, interestingly, will not work with this feature, as it's the only "Summon X" spell that is from the Necromancy rather than Conjuration school). While I'd generally advise always casting these spells at even levels, given that we get this feature at level 6, we'll look at it cast at base level.
Our Fey spirit would have 30 HP. With Durable Summons, they get 12 Temp HP, which is 40% more. Already pretty good. But that 12 THP also confers resistance to everything except Force, Necrotic, Psychic, and Radiant damage. That's admittedly a wide spread of damage types, but especially if you know what kind of threats you're likely to face, you can pick foes that are unlikely to do any of those.
An interesting rules question, though: does all the damage happen as a single instance, or do we only halve it up to the point that there's Temp HP? In other words, let's say a monster hits this thing for a crushing 30 bludgeoning damage.
Does that fully get halved to 15 total, meaning our Fey Spirit now has 27 HP? Or do we say that 24 of that damage gets halved by the Temp HP and then the remaining 6 carry over without resistance to lower the Spirit to 24 HP?
I think I'd be inclined to rule it more simply: that even if it breaks through the Temp HP, it's still all halved, and it's just subsequent hits that will no longer be reduced by resistance.
The point of this, though, is that while the Temp HP is only half our Wizard level, because it's also granting resistance, it's really (in most situations) effectively more like 4 times our Wizard level. That's quite good scaling - by level 20, our summons could take an extra 80 damage.
Indeed, I think that the main big risk of any sort of summoning spell is that there are multiple ways for foes to end it early: they can break your concentration, they can kill the summon, or they can dispel it. This one really helps you avoid or at least delay scenario number 2.
Then, at level 10, problem number 1 is more or less solved. Very simply, you can't lose concentration from taking damage.
Now, we can take feats and such to make it far harder to lose concentration. War Caster paired with Resilient: Constitution (or getting Con save proficiency some other way) is great in the majority of cases, but if you get hit with a dragon's breath at high levels that does something like 90 damage, there is no way you're ever going to make the Concentration save (though Absorb Elements can certainly help). But here, as long as they don't fully knock you out, or stun/incapacitate you, that spell isn't going anywhere.
(Note that this will help with any Conjuration spell, so if you're in "empty the tank" mode combining Conjure Minor Elementals and Scorching Ray, this will make that much safer to do so.)
Finally, at level 14, there's Split Summons. Here, when you cast a Summon spell (or any Conjuration spell that creates a creature that has a stat block in the spell - very specific here, so we're excluding things like Summon Greater Demon from Xanathar's) you can instead summon two creatures, but they each have half their usual Hit Points.
Here's the fun trick, though: they should each gain the Temp HP from Durable Summons.
So, let's say you cast Summon Fiend at 6th level, getting a pair of yugoloths (while Summon spells give you three options for what kind of thing you're summoning, I think the specification that they be the "same kind" here means you can't mix, say, a Demon and a Yugoloth).
Yugoloths created with Summon Fiend have 60 HP (the highest of the three options). That means two of them will each have 30 HP. But at level 14, we're going to be giving each 28 Temp HP. That means even before we take resistances into account, we're nearly at the same number of points of damage that the conjured creatures can take (and if we had chosen Demons, they'd only have 25 HP a piece, or if they were Devils, only 20, so they'd actually have higher HP+THP than they'd typically have without any of these subclass features). Obviously, with the resistance from Durable Summons, they wind up being significantly more resilient than a single one would be summoned by anyone else.
And then, you know, effectively double the damage of the spell.
Indeed, if we look at double Yugoloths vs. a Wyvern, assuming a +10 spell attack modifier, we're looking at hitting on a 4 or higher (85% hit chance). Each Yugoloth is making three attacks, and each is hitting for 1d8+9 (13.5 on a hit, 4.5 extra on a crit). That's 11.7 per attack and six attacks, so we're looking at 70.2 damage per turn.
Now, of course, we shouldn't compare this with our other "vs Wyvern" things because this would be four levels later, but it's clear that this feature can give us a real bombardment of attacks (the Yugoloth is one of the lighter-hitting summons, even).
Now, how you flavor a Conjurer could come in a whole lot of different varieties. Indeed, are you more excited about traveling to other planes or across great distances? Perhaps you're a bright-eyed explorer, and your summons are more about the fantastical creatures you've seen.
Alternatively, I think you can play this subclass in a more antiheroic manner. Indeed, I think when many of us think of Warlocks, we think of a practitioner of dark magic, who might conjure forth demons and other foul beings to do their bidding. Frankly, I think that the Conjurer is a better fit for that "Demonologist" or "Diabolist" archetype (it's why I thought that I might actually make my World of Warcraft Warlock a Wizard in D&D terms). While I hate to lend credibility to the Satanic Panic, doesn't it feel like we need some lower-level fiend-summoning spells in 5E?
Of course, summoning creatures need not be dark and antiheroic. Wizards get access to most of the Summon spells, and I think Fey, Constructs, Elementals, and Dragons all make for perfectly morally neutral summoning options, while Aberrations and Fiends (and Shadowspawn if we're going into 5.0 spells) give you a darker edge (and again, Undead, though that's a Necromancy spell).
Conjurers can still make ample use of the Conjure spells (though only two in the PHB are available to Wizards - Conjure Minor Elementals and Conjure Elemental,) but this subclass does really seem built around the "put a new creature on the board" options.
The new Battle Familiar spell could work ok with this subclass, but with some odd caveates: If you already have a familiar, you might not actually get the benefits of Durable Summons, as this triggers when you "summon or create" a creature with a Conjuration spell that costs a spell slot. If your DM is really strict, they might argue that because you're altering your familiar rather than conjuring them anew, they wouldn't benefit.
Also, the strict wording on how Battle Familiar interacts with Split Summons is interesting: first off, the subclass feature says you can use it when you cast a spell that summons a spirit that is included in the stat block. It does specify "spirit," which is how all the Summon spells describe their conjured creatures, even if it's not in an obvious creature type line (they're all instead titled "Aberrant Spirit" or "Elemental Spirit.") So, I think the intent is for you to use this only with "Summon" spells.
But if we are allowed to use this with Battle Familiar, we run into another interesting rules question: Battle Familiar stipulates a "one familiar at a time" rule, which means you can't cast the spell over and over to have a bunch of concentration-free familiar pets. Specifically, it says that "if you cast this spell while you already have a familiar from this spell, that familiar is replaced with the new one," and then goes on to explain how it temporarily buffs and alters an existing familiar from the Find Familiar spell.
But in theory, if we could apply Split Summons, we could cast Battle Familiar while we have zero familiar, and then immediately get two. Neither is being conjured by a new spell when we already have a Familiar.
Honestly, though, I think the specificity that Split Summons only works on spells that summon Spirits means we probably can't use it on Battle Familiar anyway. It's still a spell worth considering, as we could have yet another body on the field doing a few attacks.
Overall, though, I think that the subclass seems like a lot of fun, and one I'd be tempted to play if I hadn't spent the last four or five years (and still going) playing a Wizard already.