Green’s Dictionary of Slang

1. a nonsense term used for goodbye, esp. following a rebuff.

1934		Wodehouse Right Ho, Jeeves 49: Tinkerty-tonk. Bertie.
1939		Wodehouse Uncle Fred in the Springtime 22: ‘Can you lend me two hundred quid?’ ‘No.’ ‘Oh? Right ho. Well, in that case [...] tinkerty-tonk’.
1940		Wodehouse Eggs, Beans & Crumpets ((1951)) 97: ‘Dance on my remains?’ ‘All over them.’ Bingo drew himself up. [...] ‘Well, in that case, tinkerty-tonk’ .

2. used as a toast when drinking.

1942

T. Rattigan Flare Path I:1: Doris (over her shoulder): Gin and ginger. (To Patricia.) [...] Tinkerty-tonk, Mrs. Miller! Maudie: Tinkerty-tonk!