How can this "Bloody Brain" cocktail be made without alcohol?
Most of the things in this are very easy to substitute.

Roses lime juice is sweet and non-alcoholic (and can be found in most supermarkets). 

You can easily sub in heavy cream for Baileys: they don't taste the same, but chemically they react about the same. 

Grenadine is just a pomegranate juice simple syrup. No alcohol there at all.

The only question is what you want to sub for the strawberry vodka. Some kind of soda, most likely. It's a question of taste. It won't be easy to get something that is both fruity and clear, but you might try white cranberry juice, or white grape juice.