That depends entirely on the means by which this occours.
Your best bet is probably keeping europe cold and more densely forested, and with larger megafauna. Since neanderthals were ambush predators, while cro-magnons were persistence predators, this changes the environment enough that its neanderthals who have the competitive advantage.
When agriculture gets discovered, things might get interesting. The cro-magnons will have the initial advantage due to their warmer homeland, but i believe theres been evidence of small scale hunter-gardening by the neanderthals.
Militarily, neanderthals were bloody fearsome, we see from skeletal evidence that they were willing to pounce on ice-age megafauna, and they'd be extremely strong and tough - but their shoulder joint is a different shape that makes it worse for throwing. If they hang on long enough to see the invention of the bow that might level things out a bit.
Finally, in a sense, neanderthals have survived. There was substantial interbreeding between cro-magnons and neanderthals - I've seen surprisingly high quotes for % neanderthal DNA in europeans. I believe the red-haired gene in particular is a neanderthal gene.