Upvote:4
What might make you feel odd about these events is that you read them with a Napoleon I point of view and not Napoleon III's one.
For the first one, England was the opponent, as before for the Revolution and for the kings of France. But under Napoleon III, England is, on the contrary, a reliable ally: Even if some rivalries still exist about colonial empires, and even if some British admirals are shaken when France launches new ships, overall, the two countries are Allies:
So, France and United Kingdom were no longer opponents in the end of the 19th century. When Napoleon III had to abdicate, it was not as harsh as during the Revolution: the exile of Napoleon IV was a security for the French Republic but she did not try to kill him. So England was not annoying France by accepting him on her territory, and by enrolling in her army. On the other hand, it was not a good idea to make the son of the ex-Emperor fight in the French Army as an equal (in the Republican sense) of the other officers.