If you don't mind stretching plausibility, a mark two nuke could be used for getting around core warhead interceptors that are in core worlds, if for some reason you cannot break past the outer ring. Of course, with the strategic reserves and brutal picks, you'll still have a fight even once you break through. Some of the core eyes also got nuke immunity, if I remember correctly, neutering that advantage. Still, cutting off supply on a turtle can be promising.
A mark three nuke could be part of some crazy strategy to bumrush the AI Homeworlds with a full fleet of Fallen Spire Ships and/or Golems, since they're immune to nukes - but it would be the most impractical route to victory ever. I wouldn't mind watching an AAR where someone on a 120 system snake map used all the nuke immune Spirecraft to attempt some kind of crazy coup, though making it there before the retaliation wave would be iffy. On the other hand, Spirecraft Jumpships can hold 5/10/15/20/25 ships each, and I have loaded Spire Frigates onboard before. Without those pesky mk2-4 planets, you should be able to near immediately jump ALL of your frigates to the heart of the AI homeworld for the most insane maneuver. Bonus points for having the transceiver being built at the same time, while the Dark Spire Vengeance Generators are hanging around.
Speaking about actually practical uses of MK1 nukes: Fallen Spire Campaign.
You DO not want to plan on using a nuke, but it can be a lifesaver if you're in the middle of shard retrieval when OTHER things start going wrong, particularly if your co-op games tend towards unspoken iron man no reloads. Just... don't try to co-ordinate this without teamspeak or skype.
And most eyes can be taken out by nukes. They have the health, but not the immunity. Unless they're hanging around the homeworld.
The best way to use a mk3 doomsday nuke is to slowly build one in a co-op game, (you can afford it if you carefully manage engineers), and forget about it entirely, until a counterwave hits your forgotten system.