I'm currently working on getting 'Let It Slide'. I have 3x3 and 6x6, but for some reason 4x4 and 5x5 won't show up as done. Is there a specific way of doing them?

Have you already won the required number of games, or are you looking for them to show like 1/25 to show that they're in progress?

I already have both so can't check if it's keeping track for me, but if you've won the total number of games already, I would file a ticket and see if a UA can award them to you

Where you go to see the requirements for the achievement, 3x3 and 6x6 are confirmed as completed, but not the other two are not, even though I've done the actual puzzles. I tried the "Claim" page and have opened the link of the achievement to try to trigger the announcement, but no luck. So I thought maybe the puzzles had to be done within, like, 4 minutes and 5 minutes.

Right, if you've done the required 25 4x4 and 10 5x5 puzzles for the achievements, file a ticket and see if a UA can award them to you. There shouldn't be a time limit attached since it's not specified on the achievement page