Nods off seconds after Prime Minister's turned up (6)
I believe the answer is:
sleeps
'nods off' is the definition.
'seconds after prime minister's turned up' is the wordplay.
'seconds' becomes 's'.
'after' says to put letters next to each other.
'prime minister's' becomes 'peels' (Robert Peel).
'turned up' is a reversal indicator.
'peels' in reverse letter order is 'sleep'.
's' after 'sleep' is 'SLEEPS'.
(Other definitions for sleeps that I've seen before include "Dozes" , "Takes a siesta" , "Slumbers" , "Catches some Z's" , "Rests in the arms of Morpheus" .)