I'm always hacking mine back and it might seem a little bit bald for a while but is very forgiving and should recover. Tough as old boots is privet.

Thank you both. I'm reassured!

my husband does this to ours twice a year , more severe than this and it leafs up within 2-3 weeks. dont worry.

It is still dormant at the moment, but as soon as it starts to get a bit warmer it will green up pretty quickly, as Sbg says, Privet will shoot from quite thick branches if you cut it low enough to get to thick branches, Derek.