Yes they did but spoiler ALERT................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. Nathaniel's child and their child will have a story of their own which I couldn't find because the novel ended there tho this might not be accurate since I only mtl this for like months ago and it was google translated

I read the novel a few years ago, so this may be a little inaccurate in the detail.
Nathaniel doesn't actually love the king and it's purely obligation.
He's an omega and of a high rank and is expected to marry and have a child. When he finds out Chris takes medicine he requests it too.
He only dislikes Chris because if he has a child first, or he fails to it puts his position at risk.
He ends up with someone else he does actually love.