If properly configured the Xbox one and the PS4 Pro should be able to co-exist.
Verify that each device is configured to match your router's configuration.
However if the two combined are demanding more bandwidth than available then you will have lag.
Check that you are using Cat 5e cables from the devices to the router. And router to modem as well. Even if already Cat 5e there could still be a faulty cable slowing things down.
You might be able to do some sort of QoS adjustments to favor one device over the other. Maybe give gaming the advantage where the lagging has a greater impact versus streaming.
And do remember that the lags could be occurring once outside of your network.
Try pathping and tracert targeting the...