Hello Mitch, what is possibly likely to have happened is that the chicks all fledged recently; maybe early this morning. Being their first tentative flight some chicks do better than others; they will often fly to the nearest twig or shrub as the parents call the chicks out when its time for them to fledge - some chicks may not make it to the nearest twig/branch, etc., and may end up on the ground on their first flight attempt and I guess the cat took advantage of that fact and grabbed the chick. The chances are that all the other chicks made it out of the box to safety at height and the parents will now be tending to them and continuing to feed them for the next couple of weeks. Keep your eyes peeled and you may see the chicks around, you will often hear them before you see them as they call to be fed.
Blue Tits only have one brood per year (Great Tits occasionally have two broods) and then next January/February they start to look for more nesting sites; it is possible your box will be used again next year but could be by any blue tits or depending on the entrance size to the box other species could use it too. Once Blue Tits have raised a brood they will not breed again this year; they usually only start another brood if a first one fails very early on. To raise around 8 chicks (sometimes more) takes a huge amount of energy from these little birds and you will see at the end of raising one brood the adults, especially the female, looks very worn out and tatty ! they will moult out their feathers around August time and grow new ones.
Just to add a note; you should not clean/empty out any nest boxes until end of breeding season; I would suggest doing this Sept/October time so you can get the box back in place before the colder winter months when birds may use them to roost in .