I know the obvious out of universe reason for the discrepancy in the computing systems in the reboot universe and TOS universe in ST, but is there an in-universe account for the substantial difference in the computing systems?
The change in terminal interfaces on star ships must have occurred some time before Nero arrived in the alternate timeline. What drove the change is unknown, and will likely remain unknown. However, you can see that there are significant differences between the Original Series bridge and the bridge on the USS Kelvin.
Here we can see that the Kelvin has multiple large screens all around the bridge, view screens in front and random displays at the side/aft stations. Most controls, however, remain as individual key, switches and levers.
Later, when the USS Enterprise's bridge is revealed, although the aesthetic has changed dramatically, interface-wise the biggest change is to replace the physical switches and toggles with touch-panels. There are still some physical manipulators, even at stations that wouldn't seem to obviously need them, such as communications (see the big sliding lever right in the middle of Uhura's panel).
Thus, from the Kelvin to the Enterprise the only major change was from physical to virtual buttons, and some holographic display upgrades. Keep in mind too that it is unknown how old the USS Kelvin's bridge was, but at minimum we know that the roughly 20 years passed between the two bridge designs. In the modern world of computers that would more than account for the changes and improvements in technology (think of a laptop from 20 years ago compared to a modern one with a touch screen).
So, as I said earlier the change in computer tech started before the Narada arrived, it's cause is unknown at this time, and the cause is likely to stay unknown for the foreseeable future.
It changed because they had a bigger budget, and contemporary expectations from high tech design has changed. Looking any more into it is really nitpicking beyond realistic expectations of canon. I doubt the reboot producers/writers intended for updates to the design to be viewed as an in-universe event (you may as well ask what's the in-universe significance of Khan now being white).
But you're also comparing the wrong bridges. The Enterprise and Enterprise-A bridge has evolved throughout the franchise, even in the prime universe. Abrams clearly modeled his Enterprise on its most modern-looking incarnation from Star Trek IV:
It's somewhat similar to the Enterprise refit bridge style we begin seeing in ST:TMP:
—especially in this TMP toy set, which replaced the light grey color scheme with white:
The ST V and ST VI updates to make the bridge look more "modern" in the eyes of late 80s/early 90s audiences, ironically, makes it look much more dated today:
Anyway... back to the Abrams/ST IV bridge, here are the main similarities:
So, just like the Klingons, the tricorders, the uniforms, and the ship's exterior, the reboot designers took inspiration from the old but updated them to look more futuristic to modern audiences while still being recognizable to old school fans. And, frankly, they did a much better job with the bridge than they did with Klingons. And the bridge is also much truer to the original than the reboot engineering section.
It's entirely an art direction thing, nothing more.
If the contention is true -- that Classic Spock was sent from the 'original timeline' to this altered one, then his ship and the Romulan mining ship should bear the familiar characteristics of TNG-like starships of the time. Neither one does; the Jellyfish's controls and interior design are largely indistinguishable from 23rd century FederAbrams starships (save the insanely large viewscreen/windows/transparent aluminum/whatever-that-is for the viewable areas) and the Romulan mining vessel looks like a big wet metallic version of the Goblin King's lair from Labyrinth.
Barring a decent onscreen explanation for why this is so (the canonical status of the Countdown comics are under contention; apparently the script writers for the new Star Trek movies can't decide whether or not they actual 'count' at all) the only thing left to really surmise is that they just look different because this production crew likes their starships to look a certain way.
The effects of time travel in the Star Trek Universe are thought to have more than singular event happenings in some instances when you change the near past it may change the further past see "Year in Hell" (particularly the ending where the scientist/mad man moves away from his work delaying the time wars) and "Before and After" (which is one giant retcon of Kes's life). This is like a built in retcon.
As pointed out by Xantec and others, a lot of changes in technology were already there, before Nero arrived with the Narada.
However, there is an in-universe explanation for the dramatic technology change after the Narada/Kelvin incident (albeit non-canon), given by screenwriter Roberto Orci and director J.J. Abrams:
Quote from Memory Alpha:
The scans the Kelvin took of the Narada's 24th century technology, that went with the survivors on the shuttles, were used by 23rd century Starfleet to reverse engineer the more "advanced" technology seen in the alternate reality, according to a post by Star Trek screenwriter Roberto Orci on Ain't It Cool News. Director J.J. Abrams also said in an interview with MTV that readings from the Narada "inspired ideas and technology that wouldn't have advanced otherwise."
The AICN and MTV articles are linked on the Memory Alpha page.