Gaster's body and SOUL were spread across time and space upon him falling into the CORE, completely erasing him from Undertale, apart from Sans and Arial, since they inherited Gaster's experimented powers. Arial travels into Outertale after Undertale and began meeting up with Sans and Papyrus, accidentally telling them of Gaster when there never was a Gaster in Outertale. They misinterpreted this information as 'Gastar', and thus the theories of the Ebbot Quadrant having an external presence emerged. Dr. Alphys, in Outertale, has always been the royal scientist, and was responsible for finding the Quadrant's power supply, light-absorbing crystals, as well as creating Metatron, the TOWER, and conducting the DETERMINATION experiments.
During the Save Everyone ending, when you attempt to travel back to a past True Pacifist ending, you encounter Gaster, who you must help regain memories while dodging attacks from various battles in time.
The character Grey Kid is only person directly affiliated with Gaster in Outertale, serving a similar role to Gaster's followers in Undertale.