Go off assignation outside school (8)
I believe the answer is:
detonate
'go off' is the definition.
(I know that go off can be written as detonate)
'assignation outside school' is the wordplay.
'assignation' becomes 'date' (an assigned meeting).
'outside' means one lot of letters goes inside another (some letters go outside others).
'school' becomes 'eton' (famous UK public school).
'date' enclosing 'eton' is 'DETONATE'.
(Other definitions for detonate that I've seen before include "Trigger" , "Discharge" , "Set off explosive device" , "Cause to explode" , "let off" .)