If I recall it correctly the Manga did not have the execution scene. The King was frozen by the queen the moment she entered the throne room when he ordered to restrain Naofumi. Myne later got a slave crest to force her to tell the truth. I think she later asks him what their punishment should be and he chose exile and stripping them of everything. Plus changing Myne's name.

For Myne that would be the worse punishment. Losing her status, name, wealth, everything. Plus everyone knows what she did.

There's no telling what would happen to them after they'd be exiled.