Your paladin benefits the most, the rest is relatively unaffected
In your case, there's not that much of a big deal, primarily because you don't really have any classes that scale tremendously well with being able to quickly rest up after every fight, except for the paladin.
A 10 minute rest means you can quite literally rest after each and every fight, but that is generally not that big a deal because you will eventually run out of hit dice to use for healing. However, there are some classes which scale really well of short rests.
Some problematic class examples:
Cleric: Being able to rest after every fight means you always have your channel divinity ready.
Paladin: A paladin's channel divinity likewise comes back after short rests, and for your vengeance paladin this means that he will basically have advantage against one enemy every combat. This is a pretty big damage boost for them, which combined with fishing for smite-crits might result in your paladin being able to one-shot a lot of the important enemies if they get lucky, until their spell slots run out.
Fighter: Being able to rest after every fight means you can very quickly heal with your second wind and deal out a lot of extra damage with an action surge every fight. In addition, battlemasters will be able to use their maneuvers a whole lot more.
Warlock: Essentially infinite spellslots because you can take a quick 10 minute break to get them all back.
On the flip side, your classes don't scale that well with short rests:
Wizard: Arcane Recovery is a once per day short rest ability and your divination wizard abilities all refresh on long rests.
Bard: At level 5, he'll be able to regain all of his Bardic Inspiration on a short rest instead of a long rest, but currently at level 3, it makes zero difference.
Sorcerer: You'd be able to recover 4 sorcerery points every 10 minutes if you ever made it to level 20. Aside from that, everything refreshes on long rests.
So asides from recovering HP a bit faster, and your bard getting more bardic inspirations once he reaches level 5, the biggest problem is going to be that your paladin is going to be extra effective.
In addition, somewhere in a corner, the spell Catnap is crying itself to sleep.
On narrative time
A lot of other questions seem to claim that there is "no difference" because time is entirely narrative, but this isn't true at all. Taking that to the logical extreme, a short rest of 5 minutes and a long rest of 10 minutes should make no difference because "time is narrative" and the "DM can simply say if the rest passes or not", but this only works if your DM uses the "roll a dice to see if something happens" method of DMing.
If, instead, your DM runs the world as continuing while the players aren't involved, instead of the world "stopping" whenever the players aren't involved, then it makes a huge difference if you stop being involved for 10 minutes or for an hour. Spell durations can last through short rests, the ritual of the evil bad that takes exactly 6 hours will suddenly allow for a lot more short rests, etc.
Several official campaigns have these types of time constraints, and in a realistic setting where time keeps going when the PCs aren't there, this change makes a huge difference.