Assumed American used drugs - time to get involved (7)
I believe the answer is:
adopted
'assumed' is the definition.
('adopt' can be a synonym of 'assume')
'american used drugs time to get involved' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'used drugs' becomes 'doped' (eg athletes might dope themselves).
'time' becomes 't' (abbreviation).
'to get involved' means one lot of letters goes inside another.
'doped' placed around 't' is 'dopted'.
'a'+'dopted'='ADOPTED'
(Other definitions for adopted that I've seen before include "Took into one's family" , "Like issue raised by others" , "Chose to follow" , "Espoused" , "Taken as one's own" .)