You'd think so but internationally the rights are sold - ABC show the Sydney fireworks in Aus and the different networks have different rights in the US for the 4th July fireworks.
You would think though that if they were broadcasting in a public place and the camera just happened to drift to the sky for ten minutes after midnight there isn't much that could be done about it. Indeed they could probably sue the other for disrupting their filming of the night sky over London.
Other channels can only show a limited amount as part of the usual news rights, like how you can show the goals from a football match but not the whole thing.