Multi-source, picture-in-picture live stream
Investigate some of the software switcher products such as "V-Mix". Or some of the computer-based integrated products like NewTek "TriCaster". They will handle many more inputs than even the large (and expensive) hardware switchers like BMD ATEM, etc.  The more expensive models likely handle quite a large number of PIP images to be combined. Of course, you must have enough compatible video input ports on the computer to capture all those cameras (or whatever). But they can typically handle live inputs from many computers directly over the LAN, so that would be a big advantage for your requirement.

And many (most?) of those products also handle concurrent recording and streaming all in one solution.

Ref:


http://www.newtek.com/
http://www.vmix.com/
http://livestream.com/producer