The Pokémon's CP is kind of maintained when transferred into Let's Go. When you interact with the Pokémon, it will initially display the CP from Pokémon Go. After the Pokémon is caught, the CP will change. The new CP is derived from all the stats combined. (Source)
The Pokémon's level is maintained when transferred into Let's Go. If the Pokémon was caught via a raid, it will be lv20 (or lv25 if weather boosted) when transferred into Let's Go. (Source)
The nature is randomly generated when you capture, not transfer, the Pokémon. You can catch/reset until you obtain the desired nature. (Source).
Inspired by Ellesedil comment, you can use the Fortune Teller in Celadon City to manipluate the natures of Pokemon within the Go Park. (Source)
IVs are carried over and translated into Let’s Go equivalents. While Let's Go does not display the actual IVs, there is a formula used to calculated the IVs after transferring.
Hit Points = Stamina IV * 2 + 1
Attack & Special Attack = Attack IV * 2 + 1
Defense & Special Defense = Defense IV * 2 + 1
Speed = Randomized
The Pokémon's move set is not maintained when transferring into Let's Go. It gains four moves, and is not guaranteed to posses the move set from Pokémon Go.
Shiny forms are maintained when transferring into Let's Go.
Alolan forms are maintained when transferring into Let's Go.
Galarian forms are not supported in Let's Go. The Let's Go series supports up to Generation 7 for Alolan forms while Galarian forms are Generation 8. (Source)
Lucky status is not maintained when transferring into Let's Go. In regards to Let's Go, the luckiness only ensures the Pokémon will have high IVs. (Source)
Costumed Pokémon cannot be transferred into Let's Go. This includes:
- Sunglass Squirtle
- Hatted Pikachu
- Hatted Eevee
- Armored Mewtwo
Shadow or Purified Pokémon cannot be transferred into Let's Go. (Source)