I'm very interested in a similar setup. So far I don't think I've figured out an elegant solution.
My current solution is to have both devices plugged into the monitor, and use a usb hub to share the peripherals, manually moving the cable to the hub between devices.
Here's the exact model I ended up getting (I'm not affiliated in any way) https://www.amazon.com/UGREEN-Selector-Computers-Peripheral-...
Now I can plug my laptop into my dock, and just press a button to switch all my usb devices between machines.
It's actually a really nice workflow, nice enough that sometimes I'll leave the desktop up on my second monitor playing music or videos when I'm doing heavy processing on my work laptop, and just occasionally switch my mouse/keyboard between them with a button press.
When I want to switch, I just change the input on the monitor and press the button behind it.
It’s not the 1-click solution that I wanted, but it’s close enough and cheap.
Switching USB ports is cheap, switching video signals isn’t.
Finding a cheap switch capable of doing both is not that easy
That was on HN earlier this year. Basically it's software that you run and set up to detect when certain peripherals connect/disconnect and have it switch the input on the monitor automatically. I've been using it and, while not always perfect, it actually works pretty well, especially considering the price.
If you don't always want the monitor to switch inputs (sometimes I just want to quickly switch my keyboard/mouse to my laptop to do something but don't want to switch monitor inputs) you can also just set up shortcuts or scripts (I created an Alfred workflow, for example) and use something like https://github.com/kfix/ddcctl to switch monitor inputs.