It's questionable whether you can really call it an execution; there was no trial and there weren't even any charges. We can't call it a lynching as it there was not thing spontaneous about it and it was carried out by state actors; assassination is out for other reasons. I think that leaves us with murder. Semantics are sometimes important.
Anyway, Nikolai Alexandrovich was not a very sympathetic man and his wife was not particularly pleasant either. While it's hard to justify the straight-up political murder of a pair of (by now) harmless civilians, it's hard to feel particularly sorry either. And his last words - 'What?! What?!' - are darkly hilarious. It was wrong (and yes absolutely a crime), but barely justifies a place in the catalogue of horrors of the new Bolshevik regime.
The murder of their children and retinue though...
Well, being utterly practical, if I were in the Soviet's shoes, I would be very uneasy with the possibility of Tsarevich Alexei falling into the Kolchak's hands and all the implications.