Bird box..

Hello, 

hope you can help me.. I have a bird box that I was meaning to hang properly but due to the bad weather, I didn’t get round to it.. 

I have noticed that I have blue tits that have made a home in the box and can hear the little chirping of little ones. The box is above ground but not that high and i have just seen a local cat try to reach it.. 

Am I able to move the box to a higher place without disturbing the little ones and parents or will parents abandon the nest box.. ? 

I think the answer will be, do not love it.. but still want to ask.. ? 
thanks in advance.