The complete answer to this question is a very long one, so I will answer the more specific parts of it. First, a small correction, the Kyber memory crystal held the information on every known force-sensitive in the galaxy, as discovered by the Jedi Recruiters as part of the Acquisition Division, in addition to however else the Jedi found these things out (including happenstance, such as with Qui-gon and Anakin). On every Republic world all newborns were mandatorily tested for midi-chlorians, which raises the question as to why Palpatine wasn't discovered since Naboo was a Republic world (they joined in 868 BBY).
The Four Children
Out of the four children planned to be kidnapped for Sidious by Cad Bane in this plot as seen in TCW 2x1-3, only three of the children are named: Zinn Toa, Wee Dunn and Roo-Roo Page. Cad Bane only succeeded in kidnapping Zinn and Wee, but was intercepted trying to kidnap Roo-Roo. The other two children were eventually rescued on Mustafar by Anakin and Ahsoka before the surgical slave conditioning could be done, which Palpatine was using as a dry run for future force-sensitive lackeys as described here and more generally here, and in the following quote from the above quoted plot link.
Afterward, Darth Sidious observed the two captured children on
Mustafar via hologram with great enthusiasm. Believing that their the
Jedi would simply waste the infants' talents, Sidious took it upon
himself to mold them into agents of the dark side—a vanguard of the
Dark Lord's cadre of Force-sensitive spies and assassins. In his
vision of the future, Sidious foresaw an army of Darksiders who would
act as his eyes and ears throughout the galaxy, utilizing the dark
side to enforce his will. Hence, he commanded RO-Z67 and another nanny
droid to surgically operate on the children in order to make them
better suited to his purposes for them. The droid informed him that
such young subjects rarely survived the procedure. Sidious cared
nothing for the lives of the infants, however, and merely saw the
matter as a trial run to see how successful the procedure would be at
an early stage. Even if they died, Sidious knew that he could simply
start over by kidnapping more Force-sensitive children that he could
experiment on until the procedure was perfected. Overall, he stood
with nothing to lose and everything to gain.
I couldn't find a name for the fourth child or nothing else about Roo-Roo Page. The other two children, Zinn and Wee were not returned to their parents, but were taken by the Jedi to Coruscant to enter the Order as Initiates/younglings. Absent any other information, one may presume that they were at the Jedi Temple during Operation: Knightfall (which happened in 19 BBY, 3 years after they were kidnapped in 22 BBY) and therefore most likely perished at the hands of Vader. The whole scenario took place sometime in 22-21 BBY, though the dates are not concrete (for instance, their birth years are listed as circa 21 BBY yet Bolla Ropal and Ord Enisence died in 22 BBY, and while Ord died before the holocron theft, Bolla died after it).
The Kyber memory crystal
There is nothing that I could find in the canon to suggest that Sidious had the Kyber memory crystal, or the holocron required to open it, and I would make the following points about that:
Cad Bane stated that he did not copy the list before losing the crystal back to the Jedi.
The Kyber memory crystal was kept separate from the holocron required to open it. The holocron was kept at the Jedi Temple (hence Bane breaking in to steal it) in the Holocron Vaults and the crystal was kept by a Jedi Master, at the time Bolla Ropal. There is nothing to suggest that the Jedi altered this practice, therefore the default assumption would be that at the time of Order 66/Operation Knightfall, the Kyber memory crystal was with some unnamed/unknown Jedi and not at the Temple where it could be recovered by Vader (here and here). The canon doesn't indicate who had it, where it was or what happened to it. It is more plausible that Vader recovered the holocron used to open it, though it would be of no value without the crystal.
It is possible that the crystal was designed in such a way that it could only be opened by a Jedi, as opposed to any force-user. If any force-user could open it, it would raise the question as to why Cad Bane didn't just deliver it to Sidious or Dooku rather than going to the trouble of almost losing it in the effort to get an actual Jedi to open it. I'd also offer the following quote, which is a bit ambiguous, from the Kyber memory crystal Wookieepedia page:
The crystal was designed in such a way that its data could only be
accessed if it were to be installed in a Jedi holocron. In turn, the
holocron could only be opened by a Jedi's use of the Force.
- I could not find any mention of Luke recovering the crystal when he resurrected the (New) Jedi Order, though he did discover many artifacts and holocrons and so on.
Other younglings at the time of Knightfall
From RotS, after Vader has attacked the Jedi Temple:
Obi-wan: Not even the Younglings survived.
Yoda: Killed not by clones, this Padawan, but by a lightsaber, he was.
Obi-wan: Who? Who could have done this?
Vader slew all of the younglings/Padawans in the Jedi Temple during Operation Knightfall, but there were other younglings/Padawans on Coruscant and on other planets at the time, including those serving in the Service Corps or those who simply left the Order. Younglings that either were not selected for, or failed, their Trials and thus did not progress to become Padawans, could enter the Service Corps or leave the Order. Typically they would serve in the Agricorps. Padawans who failed their trials could also serve in the Corps. As to their fate, Sidious/Vader hunted these children down. The following quotes provide context:
The tradition ended abruptly in 19 BBY when the Dark Lord of the Sith
Darth Sidious, in his guise as Supreme Chancellor of the Republic,
ordered the Grand Army to turn on their Jedi comrades and slay them in
accordance with Executive Order 66. With the Order largely destroyed,
the newly reorganized Stormtrooper Corps of the Galactic Empire were given standing orders to capture any Padawans located during the Great
Jedi Purge and turn them over to the Inquisitorius for torture and
During the span of the Galactic Empire, Palpatine established a number
of organizations composed by Dark Jedi, including the Inquisitorius,
the Emperor's Hands and his personal Dark Side Elite. Many AgriCorps
Padawans were captured by Vader after the execution of Order 66. The
younglings that Vader did not kill (which were very few) were trained
in the ways of the dark side by several Dark Side Adepts, including
Other force-sensitives in the galaxy
This is where the answer becomes long and unwieldy because you are now talking about basically 19-23 years from 19 BBY at the end of the Clone Wars, to 0 BBY/4 ABY which was the end of Palpatine (spoiler: sort of) and the death of Vader. This period encompasses a lot of young force-sensitives who were either born right before or during these years, the most famous of which of course are Luke and Leia Skywalker, but also Galen Marek, Kyle Katarn, Mara Jade Skywalker, Shira Brie/Lumiya and a whole host of others. Sidious 'recruited' assiduously as the above links indicate, so he obviously had a method, perhaps not dissimilar from the Jedi forced-testing, to discover them, even without the Kyber memory crystal.
I haven't watched Star Wars Rebels, so I don't know how much the young Jedi are covered in there, apart from Ezra Bridger obviously, as one of the main characters, and the fact that former-Padawan Ahsoka also shows up.