the Moskva is a Fleet flagship...so I would think it would have secure unarmed nukes at all times.
Is that not a
non-sequitur? What does being a flagship have to do with carrying nukes?
It certainly carries Vulkan missiles which can be armed with nukes, but if they weren't already installed, they would probably have to be swapped over at dock rather than on-board, as it's not a quick or simple process (unscrew top, take out conventional, slide in nuclear, screw top back on!

)
Does the US follow similar procedures?
The US, since 1991/2 (Presidential Nuclear Initiatives), don't have any nukes on surface ships (or so they say!), but the Russians might still do so.
What might have happened if those missiles had struck the nuclear weapons?
You'd get an explosion (conventional) which may spread radioactive material around a bit, but I'd think it unlikely to cause a nuclear explosion. The same reason shooting a nuclear missile out of the sky probably wouldn't cause a nuclear explosion. To set off the chain-reaction requires a complex process, which any significant impact to the mechanism would likely disrupt, or at least not trigger.
Nuclear fallout rather than any actual nuclear explosion?
Yup. Radiation hazard as the nuclear material of the bomb gets scattered.
Why carry them onboard at all?
Ships are mobile, and they can get closer to their targets, and launch missiles with lower range, rather than ICBMs from places in Russia etc. I'm not saying that they do carry them any longer, but the reasoning for why they might is still valid.
Don't they have their nuclear submarines and silos to deliver retaliatory strikes if required (in self defense-not as a backup in a war of aggression or in a theatre outside their recognized borders)?
Sure, and I'm not sure any surface vessel carries nukes any more.
Do all the main militaries routinely carry nuclear weapons into battle rather than reserve them for purely defensive circumstances?
I hope not!

The US did develop the W54 "tactical nuke" back in the day, which was the smallest nuclear device they have put into service. Anything up to 1,ooo tonne yield. By contrast, Hiroshima was about 17,000. I think the intention was to use it for "demolition" of infrastructure (dams, bunkers, industrial complexes etc). So it had a backpack and everything! They may also have equipped missiles with it, but I only know of the backpack version.
