Chambers Dictionary 11th Ed.:
ORIGIN: Poss *do*llars and ca*sh*
Partridge Dictionary of Slang:
Possibly a combination of dollars and cash; there are also suggestions that the etymology leads back to doss (temporary accommodation), hence, it has been claimed, the money required to doss, or Scottish dialect doss (tobacco pouch, a purse containing something of value) – note, too, that tobacco is related to money via quid. US dosh didn’t survive but in mid-C20 UK and Australia the word was resurrected, or coincidentally recoined US, 1854
Oxford Advanced Learner’s Dictionary 8th Ed.:
1950s: of unknown origin
Oxford English Dictionary:
1953 H. Clevely Public Enemy xviii. 114 He hadn't enough dosh on him.