'Neddy' has been used in British English to mean 'donkey' - rather than horse - since the 1700s. Since about 1900, Australians have used the same word for a horse, especially a racehorse, and we may have adopted that usage.
'Gee-gee' is a child's word from the early 19th century, probably based on the instruction: "Gee up!" often said to make horses get a move on. An older use of the single word, 'gee', actually meant 'to go'.