I would like to get opinions on when to allow a character(s) to level up once achieving enough experience points.  

In my campaign, the current situation is a monk, a ranger, and rogue and a cleric are stuck in the woods on a 'camping' trip from hell.  The characters are stuck in the woods on an island that will be this environment for a while and based on the upcoming encounters they will be able to level up to level 2!  (To them exciting that they made it that far.). 

I was thinking that once the XP has been achieved that during a long rest I would allow them to level up even though stuck in the woods and far from civilization.  That seems fair, especially for the lower levels though looking at the description for long rest wonder if I need to penalize them if not resting they are focused on their upgrade.  

As I said from the start of the post what are the opinions of others when to allow character(s) to level up.  Also is there anything in Player / DMG that may talk about when leveling up can occur?

Levelling up should be an awesome positive thing. No need to penalize them for success. I personally let them level up either right when they hit it or on a long rest, and it hasn't taken anything away from gaming.

I have been a player waiting to get my cool new stuff until we get back to [wherever] and it was not my favorite. Your group isn't made up of me, so they may feel differently.

I would let them level up after a long rest where they get the full rest without any interruptions.  Which is what usually happens unless there's a story reason to keep them from leveling up for a day or two in game.

Logistically, I think its best to do at the end of a long rest. Only because so many class features, spell slots in particular, refresh then. So you don't tell the cleric they now can theoretically cast level 2 spells, but they haven't actually had the time to prepare them. And I never worry about being in town in order to level up. In some earlier editions there had been a rule, maybe it was optional, that expected characters to spend time and money to train to level up. Now I think the idea is that its more abstract and assumes characters are working to better themselves. Like the fighter had been working on timing how to swing his sword during fights, and reflecting on lessons he learned from the battles during downtime or for at least part of those eight hours of resting every day, and then one day, it finally clicked and now he has the sentinel feat.

Thank you, and everyone else for the feedback.  Since I can 'almost' map out with encounters to XP to the possible location(s) where a long rest could occur that would be the ideal place for them to level up. (All under the condition they get the XP.) I'm not going to throw any penalties as the point of making the advancement a fun occasion, as well as Oboe, logic interpreting the long rest as an activation point.

Thanks again everyone for the feedback.  I know it was a silly question but I appreciate you all taking the time to respond.

It depends on what they are doing when they level up.

No matter what, they require a full rest in an actually restful place (an inn, town, a secure camp site that is not threatened, etc). Sleeping 8 hours in the depths of a dungeon while posting watches to guard against the denizens does NOT count. The idea is that minimally, they need to think about what they have learned and spend some time figuring out how to apply it to a new situation.

However, if there is a drastic change coming with the level up, then the leveling, in addition to a "town rest" (or equivalent) requires roleplay. For example, our party got to level 2 at the end of the session before our last one. For several characters this meant a change in what they can do. For instance, the Ranger, who was technically not a spellcaster at level 1, became one at level 2. She can't just suddenly "know how to cast spells" when she has never done it before. So she RPed with the cleric, who taught her how to do Cure Wounds. Now she knows how to cast divine spells. In the future, she can just "know" new ones. But the first time, it had to be RPed. 

Similarly, the Cleric went from not being able to channel divinity and turn undead at level 1, to being able to at level 2. He can't just suddenly "know how to do it" out of the blue. He had to learn it. Luckily he had seen clerics from the local temple turn undead in the opening scene of the campaign (zombies attacked the town), so he knew it was possible and he knew they could teach him. So he went to the temple high priest and asked, the the high priest trained him.

The rogue wants to multi class in some fighter. OK, go to the Roman Century posted as the town garrison and ask for training. They have an "in" with the centurion, who has talked to them a few times, so she's willing to provide some training. It doesn't take weeks... she just shows him some moves with the gladius (short sword) and maybe how to hold his shield in a better defensive position, and explains how to shrug off injury (action surge, etc).

The point is, if you are getting some dramatically new abilities, I think you should RP the learning of those abilities (even if it is just, like our sorcerer, for IC reasons, it suddenly "happening" in combat and then you have to figure out how to control it). You shouldn't go from being able to only cast Magic Missile one day to being able, upon killing some orcs with that Magic Missile spell, suddenly knowing how to cast new spells that have nothing to do with MM. (At least, in my opinion you shouldn't.)

That's the flip side I'm seeing is how does a person justifiably get to their special abilities from 1st to 2nd when they are in the middle of a forest.  Now, at one point they will be taken to a village of Lizardfolk where they will be given a hut to rest for the night and it was that point where I could see the 'level up' occur.  I like the thoughts that you and Oboe shared with me and it gives some additional considerations I should keep in mind.  I think there is a middle ground between the two ideas and I may go with if it does not feel too forced or causes a mutiny.  I know it is whatever I want it to be it will be but I like to hear other opinions on the subject.

I think in the hut if they can actually rest and recover is reasonable... Part of the "level up in town" idea is that they can actually get to a town. If they end up stuck in the wild for 5 levels, you wouldn't make them wait across dozens of sessions and then gain the 5 levels back at town.

To me the #1 thing is that it makes in-character sense.

It also depends on the campaign you're playing and what characters your players are using.  My campaign is kind of a postapocalyptic wild-western themed game, and there really isn't supposed to be very many truly safe areas for my players to rest at, and by the nature of my game there aren't supposed to be very many hubs for them to return to, so as a DM that would be an unreasonable demand for me to have on my players (also bear in mind that I'm using milestone progression, so that does make it easier for me to control as well).  My players are also running an assassin, a wild mage, a champion fighter, and a homebrew druid.  The assassin's backstory had them training since childhood, so I don't feel they need to RP how they acquired those skills; similar story for the druid.  Wild magic needs literally ZERO explanation for how it works by virtue of it's very nature, and there is literally nothing more basic than champions in terms of class RP, soooooooo yeah.  From where I stand and where my players currently are in our game, I feel it would be unreasonable for me to require that of them.