Interesting match up — working with the assumption Nagato has haki to inflict damage on Katakuri.
Physically, Katakuri is very far above Nagato but I don’t think he’s too fast for Nagato to react to. With Kabuto in the driver seat (who logically had diminished Edo Nagato’s reaction time), Nagato was reacting to both Bee and Naruto just fine.
Ninjutsu wise, Nagato’s best options are to use the Deva Path to build distance and use the animal path for shared vision and to go aerial via the bird summoning. The only solid way Nagato is getting any attacks off is with the Chibaku Tensei or a Chou Shinra Tensei. I’d say the Gedo Statue is in play but Katakuri’s speed and future sight makes that less likely.
Asura Path might give Katakuri some problems but I don’t think it’s going to be more than a distraction. The other paths aren’t worth mentioning.
So, it basically comes down to whether or not Katakuri can kill Nagato before Chou Shinra Tensei or Chibaku Tensei. The latter? Yeah — it’s a slower technique for someone of Kata’s speed and insight. He should be able to kill Nagato before it grows into a big enough threat to kill Katakuri.
Chou Shinra Tensei? I think that’s where Kata is likely to lose. It’s a fast jutsu that’s omnidirectional and Nagato’s Six Path jutsu are > Pein’s, making his even worse for Katakuri.
It’s not impossible for Katakuri to survive it with Haki... but I don’t think he can come back after that damage. Nagato needs to only go air born and wait for his Deva Powers to return so he can use it a second time or use Chibaku Tensei.
So Nagato takes it, 7/10 just because his best jutsu is a great counter to Katakuri’s malleable body. If Katakuri ever catches Nagato he has a very good chance of murdering him, but even a lesser Shinra Tensei and the Asura Path could probably help Nagato survive CQC for a period of time.