This is extremely impressive. Someone filmed themselves using Remote Play through their phone's wireless network and it actually seemed usable.
Certainly not perfect or optimal, but the fact that it even works to this degree over a phone shows a ton of promise for using it somewhere with actual WiFi.
I believe on the PS4 stream the guys said (I may have misheard though) that you needed to be on the same connection for Remote Play to work, which Sony has stated (and this video proves) is not the case.
Actual gameplay starts around the 3:30 mark.