What the achievement asks
Complete the sixteen daily quests at the Little Scales Daycare. Ten points, account-wide, and it rewards the Axel pet.
The sixteen
The last five are the whelp races, which are also Daycare Derby.
Finish the races, do not abandon them
Daycare Derby only asks you to take part, so it is tempting to start a race, give up, and move on. Those same races are on the list above, and this achievement wants them completed.
Players have done exactly that, credited the derby, and then had to wait for the rotation to bring each race round again. Run them properly the first time.
Do them on the right character
The dailies do not appear at all until the Whelp Daycare storyline is done, and they appear for the character that did it. If you log in and the daycare has nothing for you, that is almost always the reason.
Even then the blue daily icons do not always show on the map. Walk up to each of the whelps and to Agapanthus and check them one by one rather than trusting the map.
The trick that gets you more than one a day
Doing the yellow story quest at the daycare refreshes the blue dailies. Players have watched the same three dailies they had run for three days straight be replaced with new ones the moment a yellow quest was handed in.
Do the yellow quest first, then the blues, and you will get through the sixteen faster than one a day.
How long it really takes
Sixteen dailies, offered a few at a time, gated behind the four-day cycle of each whelpling in Hey Nanny Nanny. The realistic figure players report is around nineteen days if you never miss a reset.
Where it fits
Part of A World Awoken, which rewards Taivan. This is the longest of the three daycare achievements, so treat it as the schedule and let the other two fall out of it.