This change sucks (but as a long time Withings user doesn't affect me... for now), but I haven't had trouble with either the scale or BP cuff syncing to Apple Health. I don't have any suggestions besides double-checking the Withings app has permissions to write to HealthKit. I don't see any settings in the Withings app itself besides a prompt to check "Settings > Privacy > Health > Withings App".

Sorry if this isn't helpful.

Aside: all BP cuffs and their apps sorta seem to suck when it comes to sync. I also have an Omron. At least mine measure accurately. Make sure to validate yours with your GP.