Green’s Dictionary of Slang

slingers, the n.

[SE sling out]

rejection, dismissal.

implied in get the slingers
[UK]F. Norman Dead Butler Caper 62: If he tumbles you want a lumber gaff for French lessons he’ll give you the slingers a bit lively.

In phrases

get the slingers (v.)

to be thrown out, to be dismissed from a job.

[UK]F. Norman in Vogue Oct. in Norman’s London (1969) 28: Quite a lot of charvering goes on until he gets the slingers.
[UK]F. Norman Guntz 14: I got the slingers from my job as a van driver.