I don't know the answer to your question or even if it can be done without very expensive kit BUT.
The normal way to tackle this is to record to the SD cards in your three cameras, download the files into a folder on your PC and then load them into a video editor. (Look up Corel Multicam Lite)
The video editor then aligns the three files, often using a common sound track, and then allows you to switch between the camera shots like a studio control room.
There are a number of HDMI switchers available, up to five cameras IN but one OUT to your PC (assuming your PC has a HDMI input port - mine has not) The problem is that you will only see one camera output at a time.
Or webcams. USB output and should work with Multicam Lite but no control over the webcams i.e. no zoom etc.
Corel VideoStudio Pro will give you access to 4 camera files and VS Ultimate to 6 camera files.
I'm looking to purchase 3 cameras/webcams that I can link to a PC/Mac and record at the same time.
Any recommendations on setups to capture 3 video feeds to one computer? What cameras, cords, and software?