This is based on Mining with 565 Perception (the minimum for HQ Adamantite) and the 2-4-collect 470 rotation. I expect this math translates over to Botany as well, but I haven't tried fishing yet, so don't know if it would translate over. The math also assumes that you only have one DoL at 60 whose tomes you want to buy, and/or that you can only effectively gather from one or the other. This also assumes the time conversion is 3s IRL = 1 min Eoz, which means it's 3 minutes to the hour, and 36 minutes to the twelve hour cycle. This is also going to operate on the assumption that you wait to turn Cuprite in until it's in high demand for 20% more scrips.
Each Tome of Folklore costs 99 of Rowena's Blue Gatherer's (RBG) tokens.
Each RBG token costs 50 Blue Gatherer's Scrips (BGS).
So each Tome of Folklore costs 4,950 BGS.
There are three total tomes for a given DoL, which means we have a total of 14,850 BGS to gather.
Now, the rotation above relies on Impulsive Appraisal, which has a chance at proccing Discerning Eye, which would allow us to use Single Mind to afford another collection attempt: DE-ImA-(DE/SM)-ImA-(DE/SM)-MA - yields 2-4 470c turn-ins per node. Due to the nature of it, and according to the tooltip, you have a 30% chance of proccing on either attempt. Thanks to the other mathematicians in the thread for correcting me here: since it's a 70% chance for either one not to proc, you have a 70%*70% = 49% chance for neither to proc. By the same logic, there is a 30%*30%=9% chance of both proccing. So altogether, that means 49%*2 + (51%-9%)*3 + 9%*4 = 2.6 effective turn-ins per node (eTI/n).
There is another rotation possible which guarantees 3 eTI/n (DE-MA-DE-MA-SM-MA), but it requires a slightly higher perception rating than the 565 threshold, which is only available with food, melding, or higher-level gear than the Blue Scrip set available at 58, all of which can be a substantial Gil sink if you're focusing on scrip gathering and not gathering to sell. Still, I'll do calculations for both, since it is possible some newcomers to the endgame gathering scene may be able to get that high more easily than myself (getting to 565 was a huge chore for me).
There are two paths I see for gathering the necessary scrips: the passive path and the active path.
The Passive Path
In the passive path, the player camps a single node, eliciting 30 seconds of gameplay every 36 minutes to farm their turn-ins, so they can do other stuff at the same time (e.g. work, read a book, sit in class, write forum posts, etc.). For mining, the node with the highest yield is Cuprite, which yields 72 BGS per effective turn-in (BGS/eTI).
Since we need a total of 14,850 BGS, this means we need 14,850 BGS / 72 (BGS/eTI) = 207 eTI.
With 2.6 eTI/n, we can figure out that we'll need to farm 207 eTI / 2.6 (eTI/n) = 79.615 nodes (but we'll round up to make it an even 80 nodes).
With 3 eTI/n, that becomes 207 eTI / 3 (eTI/n) = 69 nodes.
Since nodes spawn every 36 minutes (36 min/n), if you're only camping the one, that will mean 80 n * 36 min/n = 2,880 minutes = 48 hours at 565 perception = 2 days. So if you only farm a single node 24/7, it will take you 2 days total to get all the tomes for a single DoL.
But if you can manage to get that extra perception, it becomes 69 n * 36 min/n = 2,484 minutes = 41:24 hours at higher perception = 1.725 days; which we can round to 2. So a little bit more perception, by whatever means, will shave ten hours off the gathering time required, bringing us back to 2 days total. You'll just be able to sleep for a few hours before the second day is done.
The Active Path
There is a more active approach that can be taken to gathering the scrips, but it involves so much movement that you won't really be able to do anything else in between. This involves flying to both BGS-yielding timed nodes every 36 minutes. For Mining, that means Cuprite (72 BGS) and Red Quartz (54 BGS) for a total of 126 BGS per eTI (excluding Light Kidney Ore (48 BGS), because it shares a node with Cuprite).
Following the above logic, we can figure that it means 126 BGS/eTI * 2.6 eTI/n = 327.6 BGS/n (n in this case representing a node cycle).
Which means 14,850 BGS / 327.6 (BGS/n) = 46n (rounded up from 45.330).
And that translates to real time as 46n * 36 min/n = 1,656 minutes = 27:36 hours at 565 perception = 1.15 days. So it will take two days of cumulative gameplay for a player to buy all of their folklore tomes, if they are playing to maximum efficiency without spending extra Gil.
Calculating, then for the slightly higher perception, we'd get 126 BGS/eTI * 3 eTI/n = 378 BGS/n. That would take 14,850 BGS / 378 (BGS/n) = 40 nodes (up from 39.286), which would take 40n * 36 min/n = 1,440 minutes = 24 hours (one day straight) at higher perception.
Conclusion
This is probably old hat for those of you who got this out of the way early. I'm mostly posting this to get it out of my head, and to give people new to DoL some perspective. 99 RBG tokens sounds like a lot, but when you compare it to some other grinds in the game, it's not as bad as it looks on the surface. Overall it looks like a new-to-60 DoL will need to spend 2-3 days total gathering to get their first set of tomes, unless they have other DoL up, as well.
There is absolutely something to be said for increasing your perception. At 565 perception, Methodical Appraisal only hits for 112 collectability, which means you have to do more work for your collectibles, and are stuck at a lower gathering rate. However, if you can just nudge that up to 113 (with perception food/gear/materia), you gain 25% gathering rate, which can significantly cut down on the duration of the grind.
When you compare it to the DoH progression, it is absolutely a lot slower - I was able to gear up my BSM in BCS gear very quickly, and buy the Master Tome with equal haste. In my mind, the only way for this discrepancy to make sense is if the DoL classes are not going to see any further progression in future patches. Especially when DoH classes rely on DoL Folklore mats, it seems silly to me that DoL would progress so slowly by comparison, unless it was meant to last the whole expansion.
Of course, if they do give us some progression in 3.2 or onward, I hope they also implement some serious buffs to the rate at which BGS can be obtained (or nerfs to item prices). It's a really long grind, and is a huge barrier to entry for new DoLs.
EDIT: BGS are not required for progression, assuming you don't want to pay through the nose on the Market Board. Once you get 565 perception, farming Adamantite Ore for RGS is a piece of cake, and will get you to the weekly cap pretty quickly. BGS is mostly useful to support DoH classes, and to make Gil (since the materials are exclusive to those who have the tomes, and are required for DoH progression).
EDIT 2: It currently takes 26 RRG tokens plus a counterfoil to obtain the MH tool for a DoL. Depending on equipment, this could represent a substantial upgrade to Perception, which could possibly make 450c easier to achieve, and up the rate at which BGS can be obtained. I'll edit this with more details in two weeks once I've gotten my Mining MH, but for now, it seems like that might be a way to boost your BGS income.
EDIT 3: It is possible that with both Botany and Mining, your BGS income could double (assuming their unspoiled nodes aren't up at the same time, and are at least 2 Eorzea hours apart). But if you wanted to get all 12 folklore tomes, you'd also be doubling how much BGS you need, so the two would cancel out, and the time to completion would remain the same.
EDIT 4: Adjusted the math and added a paragraph to the conclusion to match what people have commented. Thanks guys!
EDIT 5: I haven't taken statistics in over a decade. Thanks to folks for providing me with the right way to calculate eTI/n when accounting for procs!
EDIT 6: Forgetting to stealth before opening an unspoiled node really sucks when there are hostile mobs around.