Battles results from the previous generation Pokémon Sword and Shield games remain accessable via the View Past Battle Records option.
While providing the first tangible links between the Pokémon HOME app and the newest generation of Pokémon games, this update expressly does not add any functionality for players to actually deposit their Pokémon from Scarlet and Violet into Pokémon HOME. The Japanese announcement states that an update to add that functionality has been scheduled for the Northern Hemisphere Spring.
Pokémon HOME is a cloud service for iOS, iPadOS, Android, and the Nintendo Switch. It serves as a complement to the Nintendo Switch core series Pokémon games, and is a successor to Pokémon Bank and the Pokémon Global Link. It was released worldwide on February 12th, 2020.