Appreciate the prompt reply, Ralph!
I have it sorted now – removed all the Gmail accounts, and then re-loaded them. That appears to have done the trick.
Not sure where the problem was, but it's pulling all through into Mail.app now, and they're all working (authentication kicked through).
Regarding Messages – the idiot in this scenario, is currently typing this post.
I had to create a new iCloud account a few months back – was suffering from terrible data consumption, with Documents & Sync burning through 4G, with the phone simply sitting on the desk (and every conceivable Wifi Only toggle triggered)... Something was corrupted somewhere, and iCloud on my phone, kept trying to call home to the mothership.
Upon the new iCloud account being created – I correctly signed in to the new account in Messages on the iPad, Mac mini and MBP – but somehow missed the most obvious one, the iPhone. Digging deeper this evening, and I only noticed now that the iPhone's Messages settings were still linked to the original iCloud account.
I signed in to the new one, lost a few weeks worth of messages, but all is now syncing as it used to! 🙂
Blushes aside, just glad I've got it working again. Many thanks regardless.