Depends whether or not Vegeta bothers to summon Nappa's pod as well. If so, Nappa comes with him to Namek and gets a Zenkai, but then his fate all depends on the Zarbon battle. I doubt he'd gain enough to be of any use against Monster Zarbon, though Zarbon would either kill him due to only needing Vegeta or spare him as well due to considering him likely to talk when pressured. If spared, Nappa escapes with Vegeta from the ship and continues to be a support character until the Freeza fight, afterwards becoming a Z-Warrior and a more relevant character than Yamcha at least. Even if he gains Super Saiyan though, I don't see him maintaining relevance after the Rosat trip due to the natural talent of Vegeta and Goku leaving him in the dust. If Zarbon kills him or if he's in a separate room from Vegeta in the ship and thus helpless against Zarbon or Freeza, nothing changes in the plot.
If he remains on Earth, however, his broken spine would leave him helpless and would be taken by Goku's friends for questioning after the battle. This would lead to them becoming aware of Freeza far earlier and being somewhat more prepared on Namek with Nappa there to inform them of various things, though likely not to the extent of changing the plot in any meaningful way. Due to the mercy of Goku's friends, Nappa would likely be swayed to their side pretty easily compared to Vegeta, though not being a combatant on the battlefield would mean he's not given a Senzu by Goku or healed by Dende, so he ends up being watched by Bulma until everyone's transported to Earth. Afterwards, the first choice of the previous scenario occurs, with Nappa at best maxing out at Piccolo tier.