I've never played one, but I'm interested.
Time limits generally scare me unless there is good walkthrough, or mechanism around them. So that is why I never got into the first three.
The next 3 are only on ps3/vita.
The first two of the third one are on Vita, with the third one being on the switch.
So I'm tempted to start there. It being on the switch, and no time limit.
But this one is also on the switch, and has little time limit. So It is tempting as well.
What you call the third one is actually the sixth set of Atelier games, because the series started during the PS1 era. But yes, the final game of that trilogy (Lydie & Suelle) is available on Switch. It has time limits on its sidequests ("Kill X number of monster type Y."), plus one during a single main quest. Lydie & Suelle on Switch is a botched port of the Vita version, so it comes with low quality assets, low resolution and framerate stutters in about half of the game's areas; a patch only addressed the resolution to make the whole thing look less blurry, but its overall technical performance remained very disappointing. If only games on Switch are considered, it's also the one where you miss out on the most story; it can still be played as standalone though.
Atelier Lulua has no time limits whatsoever from what I've played so far (about half the game). Not during its main quest nor during any of the sidequests. It's the best-looking Atelier game on Switch in both the quality of its assets and the technical performance. The ports of the first three Arland games benefit from fixed camera angles, but the very few areas with many NPCs in them show that those titles aren't clean ports because the framerate begins to stutter a bit; those rare instances shouldn't stop anyone from buying though.
As for the time limits in the original Arland trilogy...
Atelier Rorona is sliced up into twelve quarters. Each quarter demands that you finish an assignment, usually handing in a bunch of specific items. These main objectives can be conveniently cleared in 1-2 months (at times even faster), leaving plenty of time to finish at least 8 of the 12 secondary objectives (8 are enough to fill out the bingo cards that grant various bonuses that can't be earned at later dates). The time that is left can be used for whatever you consider important, such as the creation of items or finishing sidequests to earn money.
Atelier Totori's main objective is to earn enough adventurer points over the course of three years. There's not much pressure here either, because even when taking your time, you should still have about 6 months to spare because there are so many ways to earn points. Each rankup of the adventurer license is followed by a commentary on the pace you are going at, so it's possible to tell if you are doing well or if you have to speed up.
Atelier Meruru's main objective is to earn kingdom points and grow the population number over the course of three years. The game sets goals for the 1-year-mark and the 2-year-mark, so it's easy to tell if your pace is fast enough. Similar to Totori, the main objective can be finished with some months left over because there are different options to earn the points.
None of the games demand that you grind levels, so the commonly biggest time waster of JRPGs doesn't exist in the Atelier series. Equipment is much, much more important for character stats.