I read that the Queen does have to give permission in case that child one day becomes Monarch, although in this case, it would be unlikely.
The name Lilibet was apparently given to the Queen by her Grandfather as she couldn't say Elizabeth, so I don't even know if it's a proper name or just a nickname.
There has been some controversy over whether the Queen was asked if it would be ok for H&M to use this name and unless we hear it from the Queens mouth, we won't know.
Perhaps the Queen might not mind, but I wonder how she will feel if in time, lots of her subjects decide to use this name and it will become very common and not exclusive to the RF.
Up until now, if I heard the name Lilibet, I would immediately think of the Queen, but in another 5 yrs, there could be lots of little Lilibet's sitting in the classrooms around the world.