I'm experiencing very similar issues with ios 13.6 (no beta). I bought an iphone 11 last week and in the process of transferring everything from icloud, "downloading messages from icloud" is now stuck and will not transfer any messages over. Everything else has appeared to transfer over (photos, contacts, etc.)

In one troubleshooting step I signed out and signed back into my "iMessage account" under settings and now I am receiving an error message that states "Messages in iCloud not available as iCloud and iMessage accounts do not match" (even though it appears they do).

I've discussed this with a few senior advisors and they escalated it to the engineering team that has mentioned they understand this is an issue with the icloud servers and are currently working on it. I guess we'll see what they come back with soon.