The one quote given by OED is from In Strange Company, by James Greenwood, 1874, in a piece titled ”Out with the Waits”. He’s in the neighborhood of Elephant and Castle in London, and had walked to near a “St. George’s Church in the Borough” (I’m not sure where that is) with a band playing music:
“On the other hand, one inhabitant ... waxed wroth at our music. He flung up his window with a furious bang, and appearing at the opening with his nightcap on, and with a patchwork counterpane huddled over his shoulders, swore in horrible terms that if we did not that instant “sling our Daniels “ - which the Trombone informed me was a Sludge Street equivalent for moving off - he would “shy” at us every heavenly article of crockery his apartment contained.”
Maybe in the context of the band playing instruments, the phrase was referring to the instruments as “Daniels”, which would effectively mean to “put away [your instruments]”. This would make sense, especially if the local band members are used to hearing that, and interpreting “put away your instruments” as “move off”. Thus the OED may have incorrectly generalized this to any situation, with our without instruments to “sling”.
I’m not sure where “Sludge Street” is. The only one I can find in the UK is near Tuxford, and was later named Station Street or Lincoln Rd. Tuxford isn’t within walking distance of Elephant and Castle, so there must have been another Sludge Street, maybe.
It sounds like a very minimally-used local slang; I’m not sure if OED should even have included it if that one journalist’s story is the only source. I can’t find any record of the phrase or most variants in The Times (London) from 1785-1985.
Lost for Words