I can tell you that hardware solutions for this are complex, but not necessarily expensive (any more). Old video equipment can be had for a song on EBay, (look for video switchers or mixers) however, WHAT you need is complicated and bulky. I would more suggest that you go for a software solution, where there are actually lots of options. The only tech thing is how you can connect your multiple cameras to s single laptop. I think its possible if they all have usb, but again a dedicated video switcher may be necessary.
Look at all the VJ programs (Resolume is my favorite). AND its midi synchable, and has an audio component.
QUICKIE RESOLUME COMMERCIAL:
Suppose you use drum loops in your live looping, or any prerecorded loops. Just plug you looper midi out into you computer, and configure Resolume to accept midi clock. Drag your drum loops into "slots" in Resolume. Drag video clips (encoded to PhotoJPG) ont the SAME slot, now you have a SYNCHRONISED audio and video clip. Keep adding more.
Add your control surface for mixing your audio, loops, adding effects. Add a video distortion effect to the same knob as your audio distortion effect. Now you effects are synchonise visually.
Now add your camera inputs, They appear in Resolume exactly the same as a video clip, same effects, etc.
Add audio trigger effects, (split into Bas Mid Trb) have the bass trigger a "bloom" blur, have the treble trigger a waveform effect.
Noe upgrade to Resolume Avenue (I dont have this) now you get a "video sequencer" god only knows what you can do with THAT!!!
But I warn you, its addictive, and you will find yourself never doing any actual music!
Best just do what Per said, find a VJ to work with... Im looking for one!
Mark