Thank God we've moved on from the bad old days, due to the high volume of experience and research which showed all the flaws in it. God knows how we adoptive parents are supposed to help our children with their trauma and pasts if we don't know all the relevant. Our children have the right to their own story and to the facts. Of course our challenge as parents is how to talk about it
I would agree that if the child's first name has ever been in the press, then changing it is very sensible. I would not tell my child their birth surname until they were much older, because press articles and comments on them and online hate groups are vicious and not something you want your child coming across
all those people who claim to care about the poor children but actually don't give a fig about their feelings and needs 
I think talking about media involvement is going to be something which comes up later in childhood or probably teenagerhood. The earlier years are similar to other adoptions without this complicating factor, explaining the reasons for adoption at the right level for the child to understand.
A big concern with a big media case would be friends, teachers, your family, basically anyone really, finding out your child's background. Your child maybe sharing something which jogs someone's memory. You just have to be very careful what you share and with whom, and how you talk to your child about their story being private
Most media cases do fade a lot over time. Obviously we can't know what this case is, but baby p level of media interest is incredibly rare, so it's possible depending on this case, that in a few months or a couple of years, it will never be mentioned in the press again and very few people would even remember it