The question is: can sorcerer immidiately use the additional spells-per-day after level up without any rest?
100% up to the DM.
When I play, I'll assign XP at the end of the game-night but my players know they can't play leveled-up characters (at all) until they get a night's rest. I do that precisely for this reason. Players can bring two character sheets to the next session if they don't end the game with rest, and play the lower-level version until they do get rest.
Meanwhile, other DMs might hand out XP and allow characters to level immediately.
Shrug.