Whups, that was for InDesign. Illustrator is stupid when it comes to color management.
In illustrator, You have to select all of your artwork (unlock it all first,) then Edit > Edit Colors > Convert to CMYK.
Your spots will still appear in the swatch palette but are no longer connected to your art in any way. Unselect your artwork and in the swatch palette, select all unused colors and they will go away. You can then Add used colors to get your CMYK list.
This is a crap method of doing this. If you really want the more correct way of doing a CMYK conversion, at least for a conventional press, you would look up the pantone Bridge values and put them in. Or better yet, check if the conversion is even going to look the way you want it to, and if not, find a bridge color that matches and use those numbers instead.
With wide format, most machines and media are pre-profiled so that in combination they will take a spot color value and come as close as possible without doing any swaps (though if you do want an exact match, you may pay an upcharge for someone to lay eyes on it and tweek if needed.) Most wide format vendors want native files with Solid Coated spot color applied. They will take those files and apply the machine profile necessary for the media being used. You will never have those profiles in your software. You won’t know the model of the machine, the ink set used or the name of the media being fed into it. That’s all proprietary. Any place that is taking PDFs or tifs or jpgs with no bleed is not doing this step.