This question came into mind after seeing this question. It may be an absurd question, but can I use a limited-use ability which I have gained by leveling up without taking any rest? For example, can a mage use his newly gained 2nd-level spell slot right after qualifying for 3rd level? Can a fighter use his second Action Surge right after gaining his 17th level?
Much like 4e (and probably other editions), there isn't a solid guideline (As of right now) as to when XP is awarded, if XP is awarded each encounter, then, yes, you can use your new abilities right away.
However, it's not a bad idea to hold off awarding XP until you take a long rest. This keeps you from leveling in the middle of a day and having a funky situation where a wizard suddenly has more spells at his disposal (or a fighter more maneuvers) or whatever.
This is currently not covered by the existing published rules of Next and as such is currently at the DM's discretion.
I've played both ways in previous editions, and a lot depends on the flavor of the campaign world. If wizards are lone sages, constantly honing their understanding of magic with every spell they cast, then having leveling up be an "Ah Ha moment", where suddenly they just know how something works, can be appropriate. On the other hand, if you've got wizard schools in your setting, it seems appropriate to require the wizard to travel to the big city, spend some time (and money) on training, and emerge with a greater understanding. Similar arguments can be made for the other classes to require training to get to the next level in their abilities.
For most classes, leveling up mid-adventure, or even mid-encounter, is pretty straightforward. Suddenly, they're just hitting harder, sneaking more effectively, or using some awesome trick they just figured out. Magic-users and Clerics are a bit more-complicated than the general case, because of the spell slots system - if you let them level-up mid-adventure, then you need to figure out what to do with the newly-available slots that they didn't get a chance to fill at their last rest. Do they just sit empty? Do spells they've previously-expended "come back"?