The Sole Survivor can give chase immediately, and catch Ashes quickly. The Sole Survivor will still say "Go home, Ashes; Erin misses you!" despite never having met Erin or been told about Ashes. The quest will start upon interacting with Ashes.
If Ashes was caught before talking to Erin, he will be in her room. If the Sole Survivor is on Austin's tour, she will not acknowledge Ashes' presence or complete the quest until the Vault 81 Tour is over.
After doing 'Here Kitty, Kitty' Ashes will follow Erin around all the time; to her bed, to her school, etc.
If the Sole Survivor accepts the quest but does not find Ashes, Ashes will turn up inside the Institute at the reactor during the quest The Nuclear Option. He can then be sent home as normal.
Cait and X6-88 like it if the Sole Survivor kills Ashes when it escapes Vault 81.