So my party is approaching the finale of their quest, they are about a third of the way through the final scenario and they are already mostly out of slots ki etc. They already know it’s a race against time so they can’t just disengage and have a rest. Any thoughts on how they could get slots back without having a rest? 

I was thinking maybe divine intervention or something. Basically, I don’t mind if they get killed by the boss but it would be a shame if they didn’t even make it that far

Sometimes there are campaigns where you learn not to use spells.  This may be one of those learning experiences that shapes the lives of future adventurers at your table.  Are there any legitimate ways to restore spell slots? Nope.  Will they consider some kind of intervention as Deus Ex Machina? 100% Guaranteed.  You set the course.  Events played out.  They arrived at this situation and are out of resources.  

I play with one DM who is very strict on rules interpretations and we get into this kind of situation fairly frequently.  I learned in one of our earlier campaigns how valuable cantrips are in saving spell slots that I'll need later on.  Basically, I treat my spell slots like healing potions now in his campaigns.  I never use them unless I absolutely have to.  I always have both left over at the end of every major fight... because I was trained to never ever use everything or there will be dire consequences.  Oh, and the dire consequences of us not having resources? Total party kill.  After that time, I've still had total party kills with this DM, which is why I still consider it to be walking a knife edge on spending resources.  If you don't spend any, you will all die.  If you spend too much, you will all die. It's really like hardmode in D&D a lot of times, but I'm still willing to engage with that level of uncertainty because I enjoy the group that I play with and the DM actually tells some good stories, he's just demanding.

If he had brought in some kind of divine intervention or something for us, I would have felt cheated... often defeat is a better game than a victory you didn't earn.  Not in the short term... there is misery there, but also questions and discussions.  And then we tried it again with a different group in a different scenario with different tactics.  And when we win, we know we earned it.

That is not necessarily the view at all tables.  There are people that would have quit after some of our disastrous sessions.  I am stating how I have endured those scenarios and how it works at our table.  I endure those scenarios better as a player than I do as a DM as well.  It's often the hard side of the fence.  But when you put things down to a race against time, people are already expecting an outcome.  Without knowing your end result, you can have failure mean many different things.  Sometimes that is death and there is no coming back.  Sometimes it is death, but a resurrection or reincarnation (that's how my human cleric became a halfling).  Sometimes they go through a period of black and wake up in a dungeon and the world event has changed the world and they have to figure out how to overcome it from a defeated position.  Many trilogies are based on the later point of view.  Where do you want your story to go?

Thank you - maybe they’ll find an alternative route to where they need to be. There is a live option. And yes it would feel like a total fix if they magically got all their slots back.

I am fairly happy to be killing at least one of them off. An Arococra. So done with having to make allowances for flying (this is my first campaign as DM I really should have set some limits)

Oh yeah... Aarakocra can be a thorn.  Especially when the player doesn't really understand their limits and also wants to be a druid.  "You're going to fly and pick up another character?  What is your encumberance?  Oh... you're already 100 pounds over?"  And at lower levels yes... it was overpowered, but a few sniper blasts from random encounters and ours stopped just flying up at the drop of a hat to check terrain.  Eventually two other party members had flight and the monk could run circles around the 50' movement... but in lower levels... it was... well, it was what it was.  He chose it for power and never played out the in-character aspects of being an Aarakocra so... he's been punished in many other ways for those choices.

The best time to fix this would have been many, many sessions ago. A trick I've discovered is to give my players a ridiculously overpowered, one-time-use magic item. In this case it could be some kind of "Potion of Restoration" that gives the full effects of a long rest. But it could really be anything. My level 7 party has a CR 17 Solar they can summon from an Iron Flask. That should be enough to swing any one battle in their favor. A deus ex machina doesn't feel so ex machina if they feel like they earned it as a reward on a previous quest line.

An option for you now is to have the boss restore them to full strength as a sort of "fair play" move.

Or lastly, telegraph to them that they are likely to lose the battle and might have to make a hard choice to give up this objective for the chance to fight another day.

There is at least one published campaign where you encounter an (immovable) magic item that gives to the effect of a long rest. I've aalso seen the idea on a D&D game stream, in both those those it was not immediately obvuiious what they did.(I was also in a game where we were invesitgating a house and in the dining room there was a fine meal laid out. We didn't touch it convinced it was some sort of trap and the DM told us at the end had we eaten it we would have had the effects of a long rest.

I noticed you referred to them being almost out of "slots, ki etc" I don't know how much time the party (think they) have but ki is expected to be used couple after a couple of encounters so having  them discover somewhere that is clearly very rarely used (e.g. a storage room thick with dust) would enable them to have take a short rest if an hour's delay would not cause the mission to fail.

You could also have the party find a few spell scrolls which might be just enough for the boss fight while also giving hints that they still have quite a bit to get through before they encounter the boss so they need to be more frgal with resources.

The dungeon dudes (https://m.youtube.com/channel/UCQDKouT6G_6P1eBIfkTkC-w&ved=2ahUKEwiXpo_WpuvyAhUNLLkGHZKeBe8QFnoECAUQBw&usg=AOvVaw2KOUfWsN45Q68xsIV1y9Dh) have a very cool idea on their campaign, where they have this delirium potion (it's essentially a potion of liquified arcane power from a substance they have on their setting) that retrieves 1d4+1 spell slots - IIRC - but if you drink more than 1 it triggers a wild magic surge and more than that can even lead to death. I think it's very interesting to increase the adventuring day as whole, but it's introduced with drawbacks that hinder excessive usage.

You mentioned that this is your first campaign, so I assume that you’re relatively new to DMing.  How new are your players?  Are they playing this way because they don’t know any better, because you’ve used Deus ex Machina before, or both, or neither?  I started DMing early this year, and when stuff like this happens to us, I explain what went wrong and why, and give them a few options or ideas as to what they might do to survive, win, or whatever their objective is.  I only do this once, as sort of a “fair warning” kind of thing, and then after that, it’s up to them to make their plans and take care of themselves.

It wouldn’t work for every table, or maybe even most tables, but it works for us.

There's nothing to fix here.

You've given them a challenge that they accepted. They still have choices to make, they still have avenues to walk down. They *can* choose to take a rest - granted, at the risk of failure and your BBE getting released upon the land. Their other choice, to go in tattered up and risk their PCs lives in the *hope* that they can pull off some miracle.  Don't back down, they didn't. I'm with Grayfax, in that if you let up now, they will have been cheated out of all of the choices that they made to get to this point.

They will know that you pulled your punch. Let the scenario play out. 

The Catnap spell lets you take a Short Rest in 10 minutes, and there are a great many ways to either shorten the amount of time needed or take the full normal amount of time in nearly perfect safety.  An Elvin Wizard can take a Long Rest in 4 hours, and with a safe spot to do this in, gets all their spells back without having to worry much.

There are some classes, like Warlocks, that get their most essential powers back with a Short Rest.  The other spell casting classes make up for this by having vastly more spells to choose from and a much wider selection of spells.  Wizards depend on Long Rests, but have the the largest selection of arcane spells possible and once they hit 5th level, they vastly outdo melee types in every way but armor and hit points.  Clerics get the best options for Divine spells, but are limited somewhat by their domains.  Druids are even more limited, but have great utility.  The trouble with having utility is that it's situational.  Being great in combat is useful almost all the time.

What's really needed isn't so much how to give the player characters spells slots back, it's more about how you are pacing things and when you are letting people restore their resources, not just the magical kind, but even things like arrows, flasks of things to use against enemies, rations, light sources, and even small things like maps.

Unless there is some vital reason for a break-neck pace, like a time limit, why shouldn't the party get some down-time?  Even in Lord of the Rings they still had frequent Short Rests and the occasional Long Rest.

Thanks Everyone all really useful ideas. One thing I need to remember is to let them drive the story - it doesn’t have to end up how I see it in my head