Can't sign in to Apple ID or iCloud on devices - but devices show on Apple ID list of connected devices

I've recently bought a new iPad 8th Gen (A2270) to replace an aging iPad 2 (A1396). When setting up the new iPad I'm getting "There was an error connecting to iCloud" after entering my Apple ID and my password.

I have 2-factor enabled and a code does comes through on my old iPad and my iPhone ok. But this 2-factor only came up on the first attempt at signing in or again on first attempt after resetting the device.

If I log into Apple ID on a PC web browser, or look under Apple ID on either the old iPad or iPhone, I can see the new iPad in the list of devices but the new iPad is still asking me "Sign in to your iPad" in Settings.

If I skip setting up Apple ID, the new iPad can go on internet via Safari, but it cannot go to icloud.com; where it says "Connection Error - icloud encountered an error while trying to connect to the server".

New iPad is basically useless as I can't upload apps to use the iPad as the App Store needs the device to be looged into my Apple ID.

Also, as a secondary issue, both the old iPad and iPhone are now asking me to "Update Apple ID Settings" and when I try and enter the Apple ID password they come up with the error "Verification Failed - An unknown error occurred". Neither device can connect to icloud.com in their respective web-browsers - "Connection Error - icloud encountered an error while trying to connect to the server".

Seems some sort of 'handshake' between my device(s) and iCloud hasn't happened.

• Apple ID password has been changed as part of attempts to sort. Subsequent logging into Apple ID is fine from a PC web-browser, but I still have the same issues on my Apple devices.
• I've tried resetting the iPad; a few times.
• I've got a strong (3-bar) WiFi signal and fibre broadband connection to the property.
• New iPad and iPhone have been updated to iOS 14.1