Controlling regime votes against accepting new head of National Gallery (5,5)
I believe the answer is:
nanny state
'controlling regime' is the definition.
(term for an overprotective government)
'votes against accepting new head of national gallery' is the wordplay.
'votes against' becomes 'nays' (as opposed to 'ayes').
'accepting' is an insertion indicator.
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'head of' indicates taking the first letters.
'gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
The first letter of 'national' is 'n'.
'n'+'n'='nn'
'nays' going around 'nn' is 'nannys'.
'nannys'+'tate'='NANNY STATE'
(Other definitions for nanny state that I've seen before include "Interfering government" , "Undue government interference with personal choice?" , "Overprotective government deplored by Thatcher" , "Interfering authorities" .)