Disclaimer - I know that many people are not dependent on WQ gear rewards to improve their overall gear but there might be one or two gear rewards (trinkets maybe) that might be significant to some people. That in mind, my observations:
I completed a faction change about 10 days ago. Prior to the faction change, I had a gear set with iLvl 363. I routinely received rewards from WQs that were iLvl upgrades from the base reward iLvl.
Since the faction change, I got the sense that I was receiving nothing but base iLvl rewards. Because it usually wasn't terribly important that I got an upgrade (was usually just doing them because they were a convenient way to finish an Emissary), I didn't keep a log or anything. This even though my personal gear iLvl had improved slightly to 366. Monday I did 3 gear WQs back to back to back to test my assumption/gut feel and they all came out at base iLvl.
I submitted a ticket to Blizz wherein the GM acknowledged that they had received several observations of this exact nature (WQs returning to baseline iLvls after a faction change). I was told that QA was working to recreate the issue so that it could be passed on the Devs for correction.
In the meantime, anyone who is in this situation is stuck. Until QA/Devs find what has happened to break the association between character iLvl and WQ reward iLvls (and correct it), you're not going to get anything but starter iLvl gear.
The good news I suppose is that it does not seem to impact the iLvl of Emissary rewards which are still coming up for me at 370.
I doubt anyone is using WQ rewards as the deciding factor as to whether they faction change a character but, now you know.