Actually, I happen to be in a similar situation as you. My group has played games that vary in pace from Shadowrun games where we spend three real time weeks (one session per week) planning for a single run, to Pathfinder games where we clear a new dungeon every session.
Right now, we're playing a D&D 5e game set in a Dying Earth styled setting in which the world is probably going to end soon. We're all very close friends, so I think I can accurately represent some techniques the GM is doing:
Mention a prophecy. It doesn't have to have many details, or even be completely accurate, but giving the players some idea of what the end of the world is going to actually look like helps. That way, they'll know the world isn't about to end yet since they haven't seen the events of that prophecy unfold. Include several different omens, although not every omen has to be apparent right from the get-go. You might show an unexplained omen, and later have an NPC explain it to the PCs if you want the opposite effect ("oh crap the world is ending sooner than we thought").
Have NPCs prepare in a meaningful way. The players can use NPCs more well-versed in the aforementioned prophecy to gauge how close the end of the world is. In our particular example, the queen of a city-state has decreed that the citizens are to build a citadel on a particular hilltop to prepare for the end, since that's mentioned in the prophecy. The players can get a very rough idea of how close the world is to ending by knowing whether the citadel has been constructed or not; that is, the fact that an expert on the prophecy hasn't hidden away yet should mean it's still safe.
You can also take a more direct approach. If you notice that the players are getting too anxious, you can introduce a sagely NPC that feeds them more information about the prophecy, proving to them that it's not going to happen yet. In our case, we found a hermit in a myserious lighthouse in the desert, filled with clockwerk mechanisms. Turns out that this hermit has lived for perhaps hundreds of years, maintaining the beacon, and it also turns out that he's looking for a successor, because the beacon drives away aliens bent on destroying our civilization. They believe that as long as the beacon is lit, an ancient empire capable of defeating them still stands. So, in this case, the players can reasonably be assured that they're safe as long as nothing happens to the beacon.
In the case of a specific story, you have to ask what makes sense.
In terms of the mandatory 24-hour waiting period between long rests...players can always wander off and go burn the rest of the day foraging, chatting, and otherwise being idle. If they elect to take a long-rest part way through a 'dungeon crawl' make sure you consider exactly what that means for them.
For the number of encounters they can handle in a day, per the DMG, it recommends:
most adventuring parties can handle about six to eight medium or hard encounters in a day. If the adventure had more easy encounters, the adventurers can get through more. If it has more deadly encounters, they can handle fewer. (DMG84)
Remember that this is a guideline. And guidelines tend to be broken on occasion.
So, for the specific case of a published adventure, here are some considerations:
1: Look at the encounters, compare them to the XP Thresholds for encounter difficulty in the DMG. If the single dungeon (in this case, the hatchery) contains a few Easy encounters, then the party can handle it in one day, no problem.
2: Consider what makes sense. The party is raiding a hatchery that is heavily guarded. If they roll through, clear half the hatchery, then pitch a tent in a random back room...they are going to either get ambushed, or the hatchery may be evacuated while they are sitting there wasting the day. Or, at the very least, the hatchery will be on high alert and ready for them when they elect to continue...meaning ambushes, traps, fortifications, etc.
So, here are some possibilities.
First, the module may be intentionally straining your players. This happens sometimes. Forcing players to push through when they are short on resources can sometimes lead to some very creative solutions. Alternately, your players might just be burning through resources faster than they should....you really shouldn't be wasting spell slots when a pair of kobolds attacks you. If your players find themselves running short on resources, but still running into encounters, they are likely to become more conservative with their power.
Second, if necessary, allow your players to waste the rest of the day so they can take a long rest. But...consider the ramifications of them doing so. At the very least, they will put the place they are attacking on high-alert. Surprising enemies will become impossible, and I would expect the enemies to dig in and fortify their positions. They might set up ambushes ahead of the players. Or, they might decide that this location is a lost cause, and sneak out. Whatever the case may be, make sure your players know that taking a nap in the middle of an assault is probably not the best plan ever.
So, to give the general gist of it...
Yes, players can always choose to waste the rest of the day so that they can take a long rest. But you have to consider how the NPCs will respond to having a full day to discover what has been happening in their lair.
Best Answer
Adventure pieces larger than the encounter
The 5 minute adventuring day is caused by the "unit" of D&D being the encounter. You finish a unit of D&D, then you are at loose ends until the next unit.
And laying down in the middle of an encounter and having a long or short rest is an obviously bad plan.
If your "pieces" of adventure that the PCs interact with are larger than a single encounter, then laying down in the middle of that piece should also be an obviously bad plan.
The Scene
The Scene is what I call a bundle of encounters "between short rests". For whatever reason, once you start a Scene, taking a short rest somehow causes the Scene to "fail".
This can be because the Scene engaged with the PCs (and doesn't let them disengage), or it could be because the PCs learned about a problem and have not much more than an hour to solve it, or because the Scene reacts to being interacted with, and an hour break means that the Scene's reaction makes the situation worse for the PCs in whatever way.
Guards could set off alarms, or be noticed as missing. A group could respond to being attacked by picking up and moving away.
An easy, natural way to have a Scene is a relatively living dungeon, each room relatively isolated from each other, but in an hour period intrusion is noticed. Another is a running battle in the wilderness against allied foes, or in an environment where short rests are not plausible.
The breaks between encounters can be as short as 1-5 minutes in a Scene, or as long as 59 minutes. They can even be longer than an hour if short rests are otherwise implausible, like on a ship in a storm.
A good Scene budget is 2-5 medium encounters (For other difficulties Easy as 0.5, Hard as 1.5 and Deadly as 2.0). An Easy scene has 2 Medium, a Deadly scene has 5+. But adjust based on your PCs capabilities.
The Chapter
The Chapter is what I call a set of Scenes that would "fail" if you took a long rest before finishing them.
Chapters should have 2-5 Medium difficulty Scenes in them. (Easy count as 0.5, Hard 1.5, Deadly as 2.0), with again an Easy chapter is 2 Medium Scenes, Medium has 3, Hard has 4 medium scenes, and Deadly has 5+ medium Scenes.
This can be a clock (your PCs learn they have 24 hours to save the world!), the enemy responding to contact, or a difficulty in doing a long rest given the environment the Chapter is in.
From the DMG
"Chapter" and "Scene" above roughly mimics the "adventuring day" advice in the DMG, with ~2 short rests (hence ~3 scenes) and 7-9 medium encounters per day (I rounded up to 3 medium encounters per scene). It just divides it up slightly differently, and instead of using "total XP" builds things using pieces based on narrative structure. And as "sum of XP" is abstract, while "these encounters chain together" isn't, it can help build things.
The XP ratios of encounters in the DMG are 1:2:3:4 for easy:medium:hard:deadly (or pretty close). This lines up with 0.5:1:1.5:2 medium encounters ratio (I used the "medium" encounter as my unit).
Example: Raiders
An encampment of raiders. This is a Medium Chapter. It has 2 Easy and 2 Deadly scenes.
Scene 1 is a scout patrol. (Easy Scene)
Scene 2 is a heavy raiding party sent out in response to the missing scout patrol. (Easy Scene)
Scene 3 is the actual camp. (Deadly Scene)
The area has an outer ring of scout patrols, an inner ring of heavy raiders, and the camp itself. Much of the force is actually out raiding or patrolling.
In response to enemy contact they'll start getting reinforcements from raiders further out, from their home base, and eventually pull out with their treasure.
Any treasure reward is in the main camp.
Example: Travel
An eternal storm exists in a region the PCs want to cross. Crossing it requires a week of travel. There is an oasis half way. Short rests require 8 hours and require difficult checks to avoid exhaustion, and long rests are impossible.
Chapter 1:
Scene 1: Land sharks! Because everyone
hatesloves land sharks.Scene 2: Fey pranks on the wind
Scene 3: Find an Oasis. It is defended by hostiles. (LONG REST is your reward)
Note that additional hostiles will approach the Oasis if they PCs stick around. The PCs do get a day's grace.
Chapter 2:
In the event of a retreat, another hostile power could occupy the bridge, and the Oasis is again occupied by reinforcements.
Set the difficulties to taste; this is just a narrative structure.
The key here is that "5 minute adventuring day" doesn't work; the PCs would have to withdraw out of the storm, losing possibly days.
"Gritty" can make it easy for you
Use "gritty rests". In it, short rests are always overnight, and long rests require a week rest (in a safe spot).
Scenes/Chapters are tied together by plot. And plot-wise, it is far easier to justify "you fail" a Scene when you wait a day than if you wait an hour, and a Chapter "you fail" if the PCs go back to base and wait a week.
(I'd allow downtime style activities to take place during long rests).
As a bonus, this causes wall-clock time to pass, allowing the world to evolve and downtime to occur naturally. A Chapter takes 3-7 days of adventuring and 7 days of rest (10-14 days). If it takes 3 Chapters to earn enough XP to gain a level, that is a bit more than 1 month per level.
So even without a narrative break or extra downtime, it now takes about 1.5 years to reach level 20, instead of a few weeks. You can now take your sandbox and plot out events that will happen (barring PCs preventing them) on an in-game wall calendar.
Stick "T2" scale events PC can respond to ~6 months after game start, "T3" ~1 year, and "T4" ~18 months out. Have foreshadowing of these events coming before they happen (ie, if the T2 event is an overthrow of a Kingdom, hints it is happening can be dropped in month 1-5). This is a bit of a meta-clock, sort of like "Star Control 2".
This may require calibrating your encounter difficulty; Scenes now contain entire adventuring days. And single issues that are larger than a Chapter can end up taking a narrative month or more to resolve.
TL;DR
Don't build your adventure (or sandbox content!) in "encounter" pieces. Build it in larger pieces. Tie the encounters within the larger piece together.
PCs are expected to defeat the larger pieces "in detail". If they don't, they are screwed.
Consider "gritty" rests to help with narrative pacing. Justifying taking a week off having no effect is harder than doing it