Black bird nest disturbed

Blackbirds have built a nest in a very low bay leaf bush in my garden. Mum had laid 4 beautiful eggs and was happily sitting on them.  We have seen a cat lurking.  Yesterday my son noticed that the eggs were gone and the nest knocked to the side but still in the bush.  Absolutely no sign of eggs or shells.  Don't know if a cat would eat the eggs or not.  Should I now move the nest to prevent them from laying more eggs.  It really is in a very unsafe position.  Mum and dad are in garden today but not near the bush.