This means Rasputin won't be able to exert such influence as in OTL. He may still worm himself into close proximity by claiming to heal Alexei, but since Alexei isn't the heir, this is politically less critical.
Once Nicholas II dies in 1915, his son -- let's call him Nicholas III -- wil, still be quite young. He'll be 20 or perhaps even just 18. A figure who may exert influence my be his uncle, Michael, who in OTL soon became extremely despondent over the horrors of the war. Interestingly, both Olga and Tatiana served as Red Cross nurses during the war, and also came to abhor the bloodshed. If Nicholas III shares their personality, I could see him attempting to end the conflict on reasonable terms. Especially if his closest adult male relative, Michael, is strongly pushing for that as well.
If Russia takes this position, the 1916 peace proposals may find a way to succeed. Russia would be on board, the USA would be advocating for it, Karl would have taken over in Austria-Hungary and he was ardently in favour of peace...
"Home by Christmas" might come true after all, albeit two years later than expected.