Blue tit dilemma

Hello, I'll try to be concise.

I bought a new bird box some weeks ago and left it on the step outside the garden doors - with the intention of positioning it once a new fence was fixed.  Fast forward to new fence and me getting everything to hand to fix it.  However, on picking up the bird box I realised a nest had already been built in it.  I had no idea this was going on because the box is virtually at ground level and it didn't occur to me that a bird would make its nest there.

Now both blue tit parents are backwards and forwards every few minutes (presumably feeding the young?) and I feel helpless in preventing a disaster.  The neighbour's cat appears from time to time (I have a squirty water bottle handy) and I really don't know what if anything I can do to make the birds safer.

If I moved the box to a higher position, would the parents find them? Maybe hear their call?  Or would the box be abandoned?

Your suggestions would be so much appreciated. With thanks,

Jenny