I know you need an HDMI 2 cable if you want to run games in 4k/HDR, but what I want to know is if the PS4 Pro will work with an HDMI 1.4 cable when set to 1080p?
You don't, cables don't have numerical versions, only the ports at either end have.
There are however four types of HDMI cable; standard and high speed with and without Ethernet. Standard speed cables don't really exist any more because they're a bit redundant and nothing uses the Ethernet function, so as long as you've got a high speed cable that was built to spec (cheap cables aren't always compliant with the relevant standards) it'll work regardless of the resolution.
One thing that the resolution (and frame rate, HDR, etc.) does affect though is the maximum length of the cable, I've got a 5 metre cable connecting my TV to my PS4 and another for my PC and both have random intermittent blanking issues at 4K that they don't have at 1080p. That's because the data rate is much higher at 4K so the consistency of the signal is much more important otherwise the TV will get bad data and temporarily lose sync then blank out until it picks up the sync again.