@tomtom:
I think that @GraemeS has given you the best solution. The problem is that monitors that support a variety of non-USB-C / non-TB connectivity, eg HDMI and DP (or of course older monitors with DVI or VGA, which I assume are not in consideration) and also have the ability to support your USB devices are going to do so by having both the video cable and a separate USB cable “uplink” that connects into a USB port on the computer. Effectively the monitor just has a USB hub built in, no different than if you bought a USB hub and taped it to the side of the monitor.
As such, you could use a solution where you connect two cables to one computer or the other, the video cable and the cable to the USB hub, and the keyboard and mouse are connected to the USB hub. This would work, but requires connecting multiple cables each time you switch (including power if any of the computers are a laptop). Depending on where the mini is located, having to plug and unplug the cables could be nuisance, but this is workable.
A second choice is a USB-C or Thunderbolt monitor. Because USB-C and TB can both carry USB signals and video, you can get a one cable solution where you plug one cable into the USB port of the relevant computer. Since most USB-C and TB monitors deliver power as well, the laptop does not need separate power.
Unfortunately, you are still looking at plugging and unplugging from the mini in this situation. You could finesse this a bit by having two usb-c cables and leaving one plugged into the mini and disconnect from the monitor end instead and a second cable attached to the work laptop and plugged/unplugged from the monitor, but there is really nothing elegant about this solution.
Unfortunately this works better with two laptops where the plugging and unplugging is less onerous than a mini with ports that are on the back of the computer and the computer is often placed in a less accessible location (precisely because you don’t expect to be doing all of this plugging and unplugging).
You could achieve a similar result with a USB-C dock, especially if your monitor does not support USB-C or TB and you want to keep that monitor; then everything plugs into the dock, but you still have the final cable problem.
In the “olden days” there was a device called a KVM for “Keyboard Video Mouse.” It was a box with a big switch on it. You fed three connectors into it, one going to the keyboard, one to the mouse, and one to the monitor (typically in those days VGA but there were boxes for DVI as well). The box could take keyboards and mice with USB connections although more commonly the older PS2 connector. Then you ran a set of three cables for each computer to be connected into the respective ports on each computer. There were boxes that supported 2, 3, or more computers, and with three cables / computer plus the three to the devices themselves, there were a lot of wires going on, but it did work.
I do not know if there are KVM units for newer systems eg DP or HDMI for video and USB for monitors or even a USB-C switcher, but that would be what you want, I think.
There are also software solutions (eg Synergy) but with your work laptop locked down, software solutions are unlikely to be effective.