In your situation I would look into a second hand KVM switch on eBay. Some of them have both PS/2 and USB individual connectors - example below. Then you can use any USB keyboard+mouse as "master".

I have one at home that I recovered from work, thought it would be the best solution ever, but it sucks as it has a lot of input lag. I have another usb-only KVM that has close to no lag. It really depends on brands and models it seems.
In the end as I truly hate any kind of input lag and I do have the room, I have individual mice and keyboards for all my builds. But I've read from people on here that are happy with their KVM, so maybe some can give you precise model numbers that work fine.

Looks interesting! Does that allow connecting a USB keyboard to a computer that doesn't have USB, through PS/2?

Absolutely. So the picture doesn't make it obvious if you're not familiar with KVM switches. The "console" input on the front plate is where you'd connect the mouse and keyboard you'd use to control all your computers. On the back plate, the "console" mic and speakers in the top-left + the blue connector (VGA) in the bottom-left would be for your display and speakers+mic, again in a scenario where you're controlling up to 4 computers (with this model, some go up to at least 16, maybe more) using one set of screen, keyboard, mouse, speakers and microphone .

The yellow connectors + linked speakers and mic are where you'd connect to the computers you'll control through the KVM. Those are proprietary cables (afaik) but are probably not rocket science or that custom in terms of connection. On the KVM end of those cables, you have a 15 pins connector + speakers and mic, and on the computer side you get a VGA, speakers, mic + USB keyboard and USB mouse (1st set of cables) or (2nd set of cables) VGA, speakers, mic + PS/2 keyboard and PS/2 mouse.

As said before, I have a HP server KVM that handles 8 computers and I have both the USB and PS/2 version of the "client" cables, but I stopped using them after some tests as the input lag was killing me (playing Unreal Tournament with the Microsoft Intellimouse Explorer 3.0 I purchased back in the days and kept until today, there's no way I'll live with any kind of input lag :p), but as also said before, some people have had better experience with other KVM models, so YMMV and maybe some people can chime in with more positive outlooks than mine. On paper, I believe that would be the best solution to what you're trying to do.

Oh wow! Yeah the yellow connectors threw me off. I didn't look closely enough and thought they were VGA connectors, and I didn't see any PS/2 connectors. But the way you describe it sounds really ideal actually (minus the lag). I will certainly look into this. This might even allow me to get rid of one of my screens too, which would really help with the desk space issue. Except that the screen I would like to use as a single screen in that case doesn't have VGA (only HDMI, DP, and USB-C). Anyway, this is a really great suggestion, since it would also really widen my choice of keyboards.

quick reference on the kind of cables that would go with that KVM so you get the full picture:

USB being what it is, you're getting both mouse and keyboard off the single USB connector. Again, main potential issue is input lag. Second potential issue is if you're using an older mouse that is period-correct and keen on using proprietary drivers/software suite (logitech setpoint or microsoft intellimouse for example) to enable thumb mouse buttons and manage other proprietary settings, the mouse will probably not gonna be recognized anymore by the host OS once connected through the KVM. If you're using a modern USB mouse, it won't be recognized by those older drivers anyways, so no loss there, but basic function should be fine.

+1 for KVM
Although I'm going the other way PS/2 style KVM with native PS/2 keyboard and USB mouse with a PS/2 adaptor so no use re model numbers.
My enterprise 8 port KVM's by Compaq and Adderview actually perform worse then my cheap no name 4 ports. Seems like less is more.

Boohyaka is right that the mouse is only picked up as generic mouse but every mouse I've used the scroll wheel still works ok in 9x and up. Which is all I use, never went for these fancy gaming mice with loads of buttons.

Actually, it occurred to me that finding a nice compact PS/2 keyboard could be another way to go. It seems easier to adapt PS/2 to USB than the other way around (I think you can do it with just an Arduino, because the PS/2 protocol is easy to read, and the Arduino doesn't need to act as a USB host). One of the keyboards I am looking at is the BTC 5100C. Very compact, and many people seem to really like it.