Green’s Dictionary of Slang

bogbrush n.

[bog n.1 + SE brush]

a lavatory brush.

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G. Child Mixed Emotions 229: [...] referring to him as ‘Bog Brush,’ a reference to his hairdo resembling one of those spiky cleaning brushes kept beside the toilet.
P. Harper Guilt Trip 203: Wipe the toothpaste off the sink, and run the bog brush round the bowl.
C. Cocks Out of Action 54: They call him Bogbrush because his beard looks like it can get around any S-bend.

In phrases

bog brush upside down (n.)

a cropped, spiky haircut, supposedly resembling a lavatory brush.

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[UK]Partridge DSUE (8th edn).
[UK]Times Mag. 30 Apr. 50/3: I’m trying to shape up for a return to the mainstream with this functional bog brush.