It seems that Withings have done a major breaking-update that has not only broken the FitBit integration, but my VidaOne integration too.
I emailed support, and instead of just admitting that they had made a breaking change on 30th January 2019, they fobbed me off, saying they can't help me as my software (proper Windows software, not a cloud-based app that stealth-downloads changes like Alexa and every other device these days) is no longer available (i.e. unsupported).
Due to this software being old-school, it has worked absolutely flawlessly for TEN years. Right up until Wednesday 30th Jan, when I suddenly got an error message that my Withings data could not be accessed. I was able to log in, select a user, but after that, it wouldn't download my measurements. Pretty much exactly what is happening with FitBit. This is the VERY disappointing (and lazy) response I got from Withings (they did not acknowledge that they made a breaking change, even though Fitbit folks are complaining about the exact same issue):
Our Senior Support Specialist has responded from what I had previously communicated. Please know that we cannot assist with syncing issues to an app which isn't even supported by its own development team.
This, even after explaining that it is software, not an app [in today's sense ie cloud based], and therefore could not possibly have changed between Tuesday 29th (when it downloaded my weight, just as it has been doing for years) and Wednesday 30th, when it was suddenly blocked from accessing the Withings data. I mean, it's not rocket science to figure out who messed up here!
Anyway, thanks Withings, for your help. :-(