How is this difference reconciled? Is there something I missed about Nimble Escape?
It isn't. No.
It has been said before on this site (but damned if I can find it) that the CR of the monsters in the Monster Manual were not set using the methodology set forth in the Dungeon Master's Guide - there were established by eyeballing and playtest.
Notwithstanding, the DMG does say (p. 237):
Creating a monster isn't just a number-crunching exercise. The guidelines in this chapter can help you create monsters, but the only way to know whether a monster is fun is to playtest it. After seeing your monster in action, you might want to adjust the challenge rating up or down based on your experiences.
See Ogre CR calculation: is it wrong or am I missing something? for a calculation that goes the other way.
Think of the CR calculation as trying to work out how to fly a spacecraft to the moon with calculations on the back of a napkin.
This has very minimal impact on CR math
The easiest way to evaluate the impact of these changes is to evaluate how they impact CR based on the suggested guidelines for calculating CR for a custom creature.
From the Dungeon Master's Guide, Chapter 9 ("Dungeon Master's Workshop"), subsection "Creating a Monster", the following guideline is given for adjusting the CR of a creature, for Armor Class:
Now look at the Armor Class suggested for a monster of that challenge rating. If your monster's AC is at least two points higher or lower than that number, adjust the challenge rating suggested by its hit points up or down by 1 for every 2 points of difference.
Because every single creature is having its AC increased by 1 with no other changes, this is equivalent to all creatures having their Defensive CR increased by 1/2.
The guidelines for Attack Bonuses are similar:
Now look at the attack bonus suggested for a monster of that challenge rating. If your monster's attack bonus is at least two points higher or lower than that number, adjust the challenge rating suggested by its damage output up or down by 1 for every 2 points of difference.
In the same way, this means the Offensive CR of attack-roll-based creatures has decreased by 1/2. Creatures that depend on Saving Throws for most of their damage won't see an Offensive CR change.
So for all creatures, Defensive CR is being increased by 1/2, and for attack-roll creatures, Offensive CR is being decreased by 1/2, with Saving-Throw creatures seeing no reduction in Offensive CR. For the former creatures, short of especially esoteric situations, the net impact of these changes is not likely to change a creature's overall CR. For the latter, a change of 1/4CR (averaging +1/2 and +0) may be significant at lower levels of play, but is unlikely to impact anything beyond the first few levels of play.
Impact on overall game balance
Battles are going to take slightly longer than usual. All creatures are going to hit with attacks slightly less frequently, which will mean that overall, they take less damage than usual. Healing becomes slightly more powerful under this ruleset, because damage totals are lower, which means healing represents a larger proportion of damage issued during a fight.
Damage will become slightly more swingy, with characters staying at their same level of Hitpoints for longer than usual.
Players (generally) benefit from this system more than hostile creatures
This mostly boils down to the happenstance of how damage sources work. Players are more likely to depend on Spells or Saving-Throw-based Cantrips than hostile creatures are, and Saving Throws are unaffected by these changes (being already "Defender Wins" as written). So those spells will stay at the same power level, whereas weapon attacks and Attack-Roll-based spells (which are more common among hostile creatures) will be negatively affected.
Best Answer
CR is just a mechanical abstraction that doesn't exist in-universe
The CR of a given creature (a.k.a. "monster", but creature is perhaps a more generic term that sounds better for, say, a human NPC) doesn't actually tell us anything about the creature with regards to the narrative. There are other terms we use: HP, AC, XP; these terms also do not exist in-game. PCs and NPCs won't know what "HP" is. HP has an in-game description, as explained further in this question: What does HP represent?
In short, from the PHB, pg. 196:
So those are things that a PC/NPC might know about and understand; luck, the will to live, etc. They make sense in-game and are something that PCs/NPCs could talk about in-game.
However, CR doesn't even have that; it is simply a way to quantify roughly how difficult a given creature would be to defeat given it's HP, AC, abilities, etc. How that makes sense in the narrative is up to you, the DM, to decide.
Taking two examples, let's compare the Veteran and the Champion. The Veteran is a CR 3 warrior from the Monster Manual, whereas the Champion is a CR 9 warrior from Volo's Guide to Monsters. Both are just NPCs and both appear to fight like warriors (judging by their stats). The only difference between them are things that don't make any sense in-universe; their stats. Veterans only do 2 attacks via Multiattack, whereas Champions can do 3 attack via Multiattack, and has a few other abilities on top of that (such as Second Wind), as well as having higher AC and HP.
In-universe, you could justify this difference in difficulty by claiming that the Champion has had more training, more experience (I know the other is called a "veteran", but apparently this Champion has more experience), or simply that they are just naturally innately better warriors, even if they're technically not as experienced; you could even take Champion to mean that they are imbued by godly powers or something, and that's why they're more of a challenge than a Veteran. Either way, how you justify why one NPC warrior is better than another is up to you as the DM. The statblocks are just provided so that if you do decide that NPC A is better that NPC B for whatever reason, you have different stats to use for each case.
Another example is something like an Orc. Orcs are ferocious creatures to be feared, and yet they are only CR 1/2. However, other variations of orcs exist, and you could even create your own, coming up with homebrew statblocks for orcs with different training. They're still orcs, just like the CR 1/2 Orc, but their statblock would be different and would come with it's own CR, which you can use for any orcs you have decided for whatever narrative reasons should be more of a challenge than a standard Orc.
Another way to look at this would be to take a strong monsters and weaken it. The Kraken is a powerful creature with a few less powerful variants, but who's to say you couldn't come up with a kraken who isn't Juvenile or Malformed, but simply isn't as strong as the "default Kraken"? It's still a kraken, but it doesn't have as much HP or as many abilities, for whatever narrative reason you want to come up with. The only thing "CR" is interested in is how many abilities, how much HP, etc, and it doesn't care for a narrative explanation.
So, to summarise, CR makes no sense in-universe; it is just a numerical abstraction to help you, the DM, determine how much of a challenge this particular collections of stats and abilities would be (which you assign to whatever in-universe creature you decide would be represented by such stats). The fact that we usually use the Orc stats for an orc is not set in stone (you could even have an orc who uses the Champion stats, if you decided that this particular orc should be that much of a challenge...)