I got it working only after using WDRT and then reinstalling WhatsApp.
Make sure to backup the settings, text messages etc before nuking the phone.
I think I may have identified the cause of this behaviour, but I will need the help of other Insiders to confirm it.
In my case, I have found out that the phone's firmware was not the latest version, for some strange reason. I initially thought about using the Device Recovery Tool as it was suggested, and noticed the difference.
Basically, what you have to do is to get the Windows Device Recovery Tool and let it recognise your device. Your screen should look something like this:

Up to this point, there is no danger of having your phone nuked, so you can safely try this.
Now, notice the two highlighted numbers. Ideally, those two numbers should be equal to one another.
In my case, those two numbers were not equal, the phone missing at least one firmware update version.
I used the WDRT to have my firmware updated and (unfortunately, in the process) to also have my phone wiped clean. I then updated back all the way to the latest fast ring, and everything works as expected.
I'd like to ask other Insiders that have this problem to check their firmware version against the latest available. At least, this way, we can know the culprit and at least we have a solution. Not a pretty one, but a working one.