My apologies for being late to this party. It's dwelling on semantics, but Nikolai had no right to abdicate in Alexei's name, since his own abdication, several minutes before, made him, formerly Tsar of all the Russias, merely HIM, Grand Duke Nikolai Alexandrovich (or whatever title he might have had), and thus Alexei's subject. Which means then, that his abdication in Alexei's name was actually committing treason against his (not yet annointed) monarch.
Now, with that little detail out of the way, if Nikolai abdicates as OTL, his son is going to be bordering on 17/18 (unsure what the majority age was in Imperial Russia), which means this Grand Duke X (let's call him Alexander or Yuri) is on the cusp of not needing a regency. Most likely, either the Duma says 'screw it' and declares him of age to rule prematurely, or they impose a council of ministers to restrict his movements. Either way, that said, Nicky and Alicky are going to be on the first train out of Russia, they don't want their views influencing the new emperor. Knowing Alix's clingy nature, her daughters will probably be kept alongside them (or at least she'll fight for it to happen), and given Alexei's health, Nikolai will fight for that.
Then, we move onto the reign of Tsar Alexander IV/Yuri (III?). Russia's dodged a bullet with not getting Alexei as Tsar, since by all accounts, due to his health, he was petted and fussed over, spoiled, indulged by his parents and sisters. If the boy has anything of Olga's intelligence (IIRC she was labelled as one of the most intelligent of the tsar's daughters), then Russia's in for in an interesting time. He's young enough that he can still be indoctrinated with the constitutional monarchy idea, but old enough to know his own mind, most likely, he'll say 'okay, dad tried that, he's living in Paris now', and try to work within the system. (But I could also be giving him too much credit or brains for that matter. Nikolai himself was said to be like a pillow, he bore the imprint of the last to sit on him, and Russia might not be able to handle a second emperor who plays that card) - he needs to have Alexander II's brains, Pyotr Velikiy's strength of will and Ekaterina Velikaya's energy for reform (a tall order to say the least)