OK so the space-alist property is what needed changing. This should do what you want
\version "2.18.2"
\score {
\relative {
\override Score.BreakAlignment #'break-align-orders =
#(make-vector 3 '(left-edge breathing-sign clef key-cancellation key-signature ambitus time-signature staff-bar cue-clef custos))
\override Staff.KeySignature.space-alist =
#'((ambitus extra-space . 1) (time-signature extra-space . 1.15))
\override Staff.Ambitus.space-alist = #'((key-signature extra-space . 1))
\time 3/8 \key c \minor
es'8 g c
}
\layout {
\context { \Staff
\consists "Ambitus_engraver" }
}
}