Getting a computer signal to a video switcher will most likely require a scan converter. There are tons and tons available online with a huge variety of input signals and output signals. You’ll just have to find the right one to suit your setup. I’ve seen this Matrox scan converted suggested before: https://www.bhphotovideo.com/c/product/680189-REG/Matrox_CNVDVI_N_Convert_DVI.html
As far is getting your signal back to Skype, I have had success with the Blackmagic HD Decklink Extreme 3D in a Mac Pro. I ran a Panasonic HPX-170 to the SDI input on the Decklink and was able to set that as my camera in Skype. It’s not especially cost-effective, but it worked!
I know it’s not part of your question, but you might be careful sending out the full PGM to Skype since you could get an infinity loop if the full display is shown in your program. You might try sending an AUX output instead.
–Jimmy