Joachimsen would refer to "son of Joachim". This is similar to Henriksen, which would refer to "son of Henrik".
I know (Iím Danish, after all
) but you canít just use a last name that isnít legally yours. Especially not one that may be a protected name (although I donít know if Joachimsen is that).
Let's suppose he decides to study in the US. I doubt he would want to be refered to as Prince Nikolai. Also I'm ruling out the Nikolai Schleswig Holstein Sonderburg Gluecksburg...nobody in America would be able to pronounce it anyway, not to mention that it's far too long and not a surname.
But how about using the maiden name of his mother? Nikolai Manley would this be an option? Also for carrier reasons.