I am currenting working on two similar projects. One is an AT motherboard in an AT case, but using a modern ATX power supply instead. The other is an AT motherboard inside an ATX case with an ATX PSU.
For both projects, I am using this passive ATX to AT adapter, which converts the ATX power connector into an AT connector, including -5V if your ATX PSU supplies -5V (mine do). It contains two female spade connectors for the ATX soft-off connectors, with the idea being you connect those to the case's power switch. One complication is that it uses smaller width spade connectors so it won't fit a standard AT power switch.
To solve that elegantly for the AT case, I have purchased these AT power switches which have small male spade connectors on them, and have the same mounting dimensions as an AT power switch. This lets you power the ATX supply on and off using the AT case's power button, and it feels heavy when you press the button, just like the old AT switches used to.
For the ATX case, I have seen many switch variations but the particular cases I am using have those small square switches as shown in the photo above. They appear to be simply called 8x8mm switches, and here's an eBay seller offering 10 of the latching ones you'd want for $1.25. I happened to have a couple of these already from some old AT cases I cannibalised back in the early 90s (from the turbo buttons as others have stated), so I could substitute them in without any problems to turn the ATX power button into a latching switch. I have yet to decide how to connect these to the ATX soft-off spade connectors, as I don't have any small male spade connectors handy and I'm not so keen on cutting them off in case one day I want to move the converter somewhere else. I'll probably make some kind of spade-to-0.1-inch connector, since the replacement latching switch already had cable with a "TURBO SW" connector on the end of it.
One thing worth noting is that the soft-off wires in the ATX-to-AT converter are only 21cm long, which is fine in my AT cases, but it would be too short to reach the power switch in the ATX ones - so probably just as well I need to build a converter!