I would say "Nothing". Alexei was 9 y.o. very sick youngster, who didn't have a popular party or some uniform group among old elites to back him. Besides, despite Bolshevik carnage, enough cadet branches of Romanovs did survive to head hypothetical monarchist revival movement, would it be any appetite for tsarism reloaded among Russian populace. The mere fact that monarchism did not become significant force IOTL (monarchists were but one of fractions among Whites, and not a major one) places a huge heavy cast-iron lid on any "monarchist miracle" TL.