Crafty swallows went over island fast (8)

I believe the answer is:

speedily

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'fast' is the definition.
(I know that fast can be written as speedily)

'crafty swallows went over island' is the wordplay.
'crafty' becomes 'sly' (sly can mean crafty or cunning).
'swallows' is an insertion indicator.
'went over' becomes 'peed' (both can mean urinated. I am not sure about the 'over' bit.).
'island' becomes 'i' (geographical abbreviation).
'peed'+'i'='peedi'
'sly' going around 'peedi' is 'SPEEDILY'.

Can you help me to learn more?

(Other definitions for speedily that I've seen before include "Quickly, rapidly" , "In haste" .)

I've seen this clue in The Guardian.

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