You must be using TeleportService for the teleportation, if you did not know Teleportation Service has a 4th parameter which is “customLoadingScreen” you can use the parameter to set your custom loading screen and then in the 2nd game(where you are teleporting to) you can add a script into ReplicatedFirst and you can remove the default loading screen and then do local gui = TeleportService:GetArrivingTeleportGui. You can code it so if its not nil which means there is custom loading screen, you set the gui parent to playergui.
Edit: Actually, you can use TeleportService:SetTeleportGui(yourGui) just before you are using the teleport line.