The nest wifi should update itself automatically as new versions are released. Would suggest rebooting both the router as well as your modem to see if that clears up the issue. If it does not, test a hard wired device directly from the modem to see if the speed is slow right from the internet provider, it could be an provider issue as well
It's not a desktop app it is only for mobile devices so yes you would run it on your iPhone or iPad. The speed test will measure your networks performance. Since your iPad is working well on wifi, I suspect the performance issue is actually your Mac mini.
Can you tell me the results of your speed test on your iPad..and the same result on your Mac mini?