I can't speak to all times and places, but I can speak to sixteenth-century England. Briefly, no, naming wasn't delayed due to high infant mortality. On the contrary, the high risk of death led to babies being named as early as possible.

Infant mortality was in fact horrifyingly high. It's hard to get an exact reliable figure, but most estimates I've seen hover around 30%. Baptism was considered necessary for a person's soul to enter heaven, so the high possibility of a baby dying made it extra important for it to be baptised as soon as possible. The 1559 Book of Common Prayer says:

The Pastours and Curates shal oft admonish the people, that they deferre not the Baptisme of enfantes any longer then the Sonday, or other holy day, next after the childe be borne unlesse upon a great and reasonable cause declared to the Curate, and by him approved.

It also says:

Nevertheles (if necessitie so require) children may at al tymes be Baptized at home.

In other words, if the baby seemed like it might die before the next Sunday or holy day, you could baptise it at home any time. Just get that baby baptised.

And you can't baptise a baby without giving it a name. The Book of Common Prayer specifies the point at which the priest will use the baby's name in the baptismal ceremony - 'N' here is to be replaced by the baby's name:

N. I Baptize the in the name of the Father, and of the sonne, and of the holy Ghost. Amen.

So, at least in sixteenth-century England, a baby was overwhelmingly likely to be named before it was a week old.