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.

The basic principle relies on the acid (lime) and alcohol curdling the cream in the shot. You could do something similar using cream and any acid. I'm assuming you're making this for kids (I cannot imagine any adult drinking something this gross without the benefit of an alcoholic buzz)? You could try cream and apple juice, might be acidic enough.

Or you could go full on crazy with it. At a wild guess, you could make cream 'noodles' using liquid nitrogen, then drop the noodles into a warm alginate bath. The alginate should react with the calcium in the cream to encapsulate the noodles, and the freezing should allow them to maintain their shape while the chemical reaction takes place. Then add to whatever liquid you desire. Alternate options would include setting noodles with agar or gelatine mixed with cream, then placing into $liquid that has had xanthan gum added to suspend the noodles.