A creature can be grappled by multiple creatures, and it has to make a separate escape attempt against each one.
Grappled
A grappled creature’s speed becomes 0, and it can’t
benefit from any bonus to its speed.
The condition ends if the grappler is incapacitated
(see the condition).
The condition also ends if an effect removes the
grappled creature from the reach of the grappler or
grappling effect, such as when a creature is hurled
away by the
thunderwave
spell.
There's nothing in the Grappled condition that would prevent multiple creatures from grappling a single target.
Where it gets complicated, though, is breaking free. From the grapple rules for monsters, we have this:
A creature grappled by the monster can use its action to try
to escape. To do so, it must succeed on a Strength (Athletics)
or Dexterity (Acrobatics) check against the escape DC in the
monster’s stat block. If no escape DC is given, assume the
DC is 10 + the monster’s Strength (Athletics) modifier.
This says that the creature must succeed on a check against the escape DC in the monster's stat block. That indicates that it's talking about a single escape DC, a single monster, and therefore, a single escape attempt.
So, this makes it clear how to deal with different creatures with different escape DCs - the grappled creature has to make a separate escape attempt for each one.
All the creatures you've listed have grapples that work exactly the same way, just with slightly different numbers. So, as above, the grappled creature would have to make a separate escape attempt against each escape DC.
This is a fairly powerful tactic at low levels, but against stronger creatures the initial grapples are unlikely to succeed, and higher level enemies often have ways of dealing with grapplers (like teleporting out, or just killing them all at once). It's also important to remember that you can't grapple a creature more than one size category larger than yourself, so your snakes will never be able to grapple a tarrasque (for example).
With all that said, if this tactic is becoming a problem, the DM might consider allowing a monster to burst out of multiple grapples at once, especially if it's clearly strong enough to do so.
Alternative Solution: The enemies don't matter
You said that
My goal is to make coming out of the dungeon a frantic race against time.
Therefore, it doesn't really matter how many skeletons you pitch against them, as long as it feels like a battle they can't win. This is a place where the numbers are irrelevant and it's more about creating a compelling narrative.
Side note, if you haven't yet read The Angry GM's How to Build F$&%ing Awesome Encounters! (warnings for inappropriate language) you may find it an excellent thought exercise given that you say your group leans towards a roleplay-heavy style. I'm going to be using his guide for inspiration here.
The Mission
So you want to build an encounter. Before anything else, you're going to need a hook. Luckily yours is pretty easy to isolate; they've won their big baddie fight, and they now need to leave. Preferably alive.
The Mission: Get out.
But there's a complication; the longer they stay down there, the more skeletons come alive and try to murder them. It's tempting to say that our mission is then
Get out without being killed by skeletons.
But there's an interesting point to be made here; the danger they're facing isn't actually skeletons, it's time. It doesn't matter how many skeletons they do or don't fight, their chances of survival get slimmer the longer they stay in the crypts.
So let's revise our mission now that we know the danger:
The Mission: Get out quickly.
We also have the setting; a series of winding crypts that they may or may not remember how to get out of. They're going from Point A (the boss' chambers) to Point B (the exit) and need to do so as quickly and efficiently as possible.
To me, this no longer sounds like a fight encounter, because fighting isn't the mission here. Instead, I would treat this like a chase, which is going to play out very differently and likely feel a lot more urgent.
Planning a Chase
The Angry GM does a much better job of explaining this in detail than I ever could, but here's my quick take on it. I've actually run two different "chase" scenes via this method and my players loved it. They were the kind of sessions they were still talking about next week, which is always a good sign!
1. Choose Decision Points
You know your players are going from Point A to Point B. Decide how many decision points you want between those two places. Each decision point is a place where your players can choose to do something that may help or hinder their escape.
Maybe there's a closed door they need to open, a pit they need to cross, a trap they need to disable, and a dark place where they can't see. You've now made a 4-decision-point trip from A to B. At each of these hurdles your players will need to make snap decisions as a team to get everyone through.
If you were to "map" this out it might look like
Boss - door - pit - trap - dark - Exit
Each of these points should have its own "skill challenge" applied to it. I'll explain what I mean by that later.
2. Add your baddies
There has to be a reason why they can't just take their time through your decisions, otherwise the chase loses all meaning. Luckily, you already have this reason; skeletons!
For the sake of this encounter, it seems logical that a party moving at average speed can cover the same amount of distance as an army of skeletons can. Therefore, both groups will progress at exactly the same speed, neither gaining nor losing ground, as long as nothing is done to accelerate or impede this progress.
This gives us a unit of time to work with; in one "round" the skeletons can move one "space".
3. Plan your skill challenges
You now know that the skeletons will progress one space per round. Now you can decide how many rounds your players should need per decision point, where you can allow them to gain or lose spaces, and how hot on their heels you want your skeletons to be.
This sounds like a lot to plan, so let me show you just one single decision point as an example:
At the start, your "map" looks like this:
hallway || hallway || hallway || locked door || hallway
Aka, there are three spaces of hallway before your locked door that your party needs to get around. Let's say the party has gotten to the door, and they can hear an encroaching skeleton hoard behind them.
(skeletons) hallway || hallway || hallway || (party) locked door || hallway
For each thing they try to get the door open, the skeletons will advance a space. If the skeletons get to the same space as the party is occupying, the party will die. (Or, if you don't want it to be a pass/fail mission, maybe they just need to fight a fairly hard encounter to win back ground and keep moving.)
Your rogue tries to pick it with lockpicks. He fails his skill check.
hallway || (skeletons) hallway || hallway || (party) locked door || hallway
Your wizard tries to burn it with magic. He fails his skill check.
hallway || hallway || (skeletons) hallway || (party) locked door || hallway
Your fighter tries to bash it down. She succeeds her skill check!
hallway || hallway || (skeletons) hallway || open door || (party) hallway
The party bursts forward into the hallway as the horrible clicking noises of the skeletons grows louder behind them...
Trust me, if you narrate that well and make sure no one dawdles while thinking for too long you'll have your players on the edge of their seats!
Ideally your skill challenges will offer more varied solutions than just "defeat door", but this is something to tailor to your party. And speaking of...
4. Offer "Shortcuts"
Decision making isn't fun if there's only one choice. Split your crypt up into multiple paths, let your members make perception, investigation, survival and other applicable checks to find shortcuts or workarounds for some members but not others.
Also, allow your party to impede the skeleton's progress, maybe with an acid spill on the floor or by caving in a bit of the ceiling. Each time they come up with something clever, don't have the skeletons gain on them that round.
As an example, in one chase I ran there was a narrow alley that our ranger could fit in but our fighter couldn't - the ranger ran ahead and kept sending Message back to the fighter with tips about what was coming. I gave the fighter advantage on her rolls since she had been informed what was coming up. Stuff like this gives each member a chance to shine for their own skill set.
5. Pulling it all together
Once you've mapped out all your decision points and skill challenges you can make yourself a little cheat-map with each space and what will happen in it. Whip up a little token for your party and another for the skeletons and keep track of when the party does something clever and gains a space, or fails a challenge and the skeletons get closer. This doesn't have to be something your players see, just use it to keep your narrative accurate.
Then it's all up to your narration and how quickly your players can think on their feet! If they play their cards right and are clever about it, they'll never actually have to fight a skeleton and you will have built an amazing encounter!
I can't believe I found this but I actually have a picture of one of the encounters I built.
The left sheet has my "spaces" mapped out so I can track how far ahead/behind my players each are.
The right sheet has the decision points listed (in this case right-to-left) with notes to me about where shortcuts are and what skill challenges are needed.
My players never saw any of this, it was just info so I knew what was happening and could narrate accordingly.
Best Answer
Use scouts
Digging and building mineshafts is very easy to detect for even a casual observer. The BBEG can wait until the druid has wasted a bunch of his spells, then show up and bury the intruders in the grave they just dug. If it's an ambush, the PCs won't be able to teleport away without taking a few hits, and if they have a reputation for being flighty, the BBEG is likely to use poisons, diseases, curses, and other ongoing problems that won't be solved by a quick rest.
Use consequences for a slow approach
When the BBEG sees the PCs digging, he can predict where they'll end up and trap/reinforce that area of the lair. The PCs break into the lair, and it's flooded with neurotoxin or acid. This goes double if the PCs run away and then try to come back - the area will be swarming with minions, there will be defenses erected, and so forth. The BBEG may send his own spies out to find the PCs' base (or things they care about) and issue a retributive strike.
Use their reputation against them
Digging big ol' holes in the ground is far from business as usual in the realm of fantasy warfare. Stories of their weird tunneler types will spread, and evil overlords everywhere will defend their bases against this approach. The simplest solution is to bury explosive devices in multiple layers, so that the digging animals are killed and the druid needs to use another spell.
Use special terrain
Digging in dirt is easy. Digging through mud is easy, but maintaining the shaft afterwards is hard. Digging through granite is all but impossible. Digging through lava gets you dead. Digging into another plane or into a sky fortress makes no sense. If there's an underground river, digging underneath it becomes a challenge - you need to keep all that water out.
Use subterranean monsters and lairs
All those badgers are coming from somewhere. Some of them may have already done some digging - and instead of landing into the BBEG's lair, the party lands into a giant badger warren, or a fomorian hive, or the Underdark.
Use skill checks
Mine-shaft structuring is a complicated business; in real life, it's a job people go to college for. Does your druid have structural engineering knowledge? No? Then he put the support in the wrong place and the mine shaft collapsed.
Use third parties
There are more people in the world than the BBEG and the PCs. What if the PCs are digging in the middle of a sacred elven grove, and the elves come and ask them to stop? What if all their digging attracted another adventuring party, who decided to storm the citadel the old fashioned way and has already cleaned up by the time the PCs break in? What if the PCs are digging in crown lands, and a magistrate comes and demands to see their zoning permit?