Honestly, I think a big show trial ending in an execution of Nicholas would be a net positive for the Bolsheviks. International opinion might not be impressed, and certainly some people in Russia wouldn't like it, but among the people the Bolsheviks needed to win over (former SRs, peasants, minority nationalities), there was very little love lost for him. Any opportunity the Bolsheviks have to remind everyone of all the horrible crimes he committed and position themselves as the ones taking him to task for it is going to benefit them.
If there's no risk of them escaping and being used as White figureheads, I don't see the kids being executed. Lenin was well aware of how it would look to hand down a public death sentence to a bunch of kids who hadn't actually committed any crime. I think they're probably taken out of the public eye and subjected to a lot of reeducation efforts (a la Puyi). Maybe they get purged later, but that stays far out of the spotlight.