CP
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)
Level
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)
Nature
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
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
(Source)
Moves
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
Shiny forms are maintained when transferring into Let's Go.
Alolan
Alolan forms are maintained when transferring into Let's Go.
Galarian
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
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
Costumed Pokémon cannot be transferred into Let's Go. This includes:
- Sunglass Squirtle
- Hatted Pikachu
- Hatted Eevee
- Armored Mewtwo
Shadow/Purified
Shadow or Purified Pokémon cannot be transferred into Let's Go. (Source)
Yes, Let's Go Pikachu and Let's Go Eevee each have exclusive Pokémon that will allow you to catch1 in each version. According to this guide, the exclusives are:
Let's Go Pikachu
Oddish, Gloom, Vileplume, Sandshrew, Sandslash, Growlithe, Arcanine, Grimer, Muk, Scyther, Mankey, Primeape
Let's Go Eevee
Bellsprout, Weepinbell, Victreebell, Vulpix, Ninetales, Meowth, Persian, Koffing, Weezing, Ekans, Arbok, Pinsir
1. In Let's Go Pikachu, I was able to trade 5 Growliths for a Persian in Vermilion City. In Let's Go Eevee, a friend of mine was able to trade 5 Meowths for an Arcanine. I don't know all of these instances, but I do know of this exception.
Best Answer
The good thing is is that your save game is stored on your Switch itself (as explained in this answer). Therefor, you can buy/rent/borrow a different cardridge of Let's Go Pikachu to access that savegame.
The bad news, is that you cannot transfer Pokemon to Shield/Sword yet. A new service, Pokemon Home, is not released yet. When Home is coming out, it should be possible. It is set to release 'Early 2020', so you've got some time to prepare.