Cassell's Slang has
toodle-/tootle-oo phr. [20C] goodbye (cf. TOODLE-PIP; TOOROO) [etymology: ? Standard English toot, the tooting of a horn, in this case as a coach moves off, or French à toute à l'heure, goodbye]toodle-pip phr. [1920s-30s] a nonsense word used to say goodbye (cf. TOODLE-OO)