One would think that the fact that she was born early and fragile would make her baptism more urgent... not postponed until later. Why take such a risk of God forbid...the baby dying unchristened?

But in all fairness to Athena's parents, I am thinking like a Catholic, with a Catholic understanding of the importance of the Sacrament of Baptism.
It might be different from their own.
Interesting question, thank you. Of course, I understand that while Princess Beatrice and Edo Mapelli Mozzi are both Anglicans, that does not necessarily mean they are devout or strictly observe all the principles of their church, but nevertheless it is interesting to see what the church canons have to say.
The Book of Common Prayer (which remains officially incorporated in the Church of England canons, even though the usage of its liturgy has been mostly supplanted by the newer Common Worship liturgy) begins the infant baptism service in these words:
“Dearly beloved, forasmuch as all men are conceived and born in sin, and that our Saviour Christ saith,
none can enter into the kingdom of God, except he be regenerate and born anew of Water and of the Holy Ghost [...]”
Public Baptism of Infants, from The Book of Common Prayer (1662). Cambridge University Press, 2006 edition.
www.churchofengland.org
To my reading, this implies baptism is a precondition for the infant to “enter into the Kingdom of God”. (Does that mean becoming a Christian, or entering heaven after death?)
However, the Church of England’s instructions for “emergency baptism” state:
“2 Parents are responsible for requesting emergency baptism for an infant.
They should be assured that questions of ultimate salvation or of the provision of a Christian funeral for an infant who dies
do not depend upon whether or not the child has been baptized.”
Baptism and Confirmation from Common Worship: Christian Initiation by The Church of England.
www.churchofengland.org
This seems to imply an infant who dies may achieve “ultimate salvation” even without being baptized.
I am not informed enough about Christian theology to understand the difference between “entering the kingdom of God” and “ultimate salvation”.
Regardless, the Church of England canons do treat infant baptism as an urgent matter not to be “delayed”, especially when the infant is dying:
“B 22 Of the baptism of infants
[...]
2. If the minister shall refuse or unduly delay to baptize any such infant, the parents or guardians may apply to the bishop of the diocese, who shall, after consultation with the minister, give such directions as he thinks fit.
[...]
4. No minister shall refuse or, save for the purpose of preparing or instructing the parents or guardians or godparents, delay to baptize any infant within his cure that is brought to the church to be baptized, provided that due notice has been given and the provisions relating to godparents in these Canons are observed.
[...]
6. No minister being informed of the weakness or danger of death of any infant within his cure and therefore desired to go to baptize the same shall either refuse or delay to do so.”
Section B of the Canons of the Church of England
www.churchofengland.org
For what it is worth, the Mail article "understands" that the York family scandal was the reason for Athena's delayed baptism.