I don't think the writer of this encyclopaedia article was correct in reference to the Church of England. Though the Church of England had split from Rome, a lot of practice and belief and theology continued much as before . (But in English. This was the complaint of the puritans).The ceremony of Churching is still in the prayerbook and was quite common until the latter part of the 20th C.
At this period (1600s in the question) and undeed until much later non baptised babies (and suicides) would not be buried in consecrated ground. They would not appear in the burial register .
Many were baptised very close to death (and who judges when death occurred?) We have evidence of this.Most registers have large numbers of babies 'privately baptised'. A few even say that the vicar went to the house because the baby was sick. Midwives were given instructions on how to baptise and had a duty to do so.Sometimes when they expected the baby would be born dead, they baptised it before it was fully delivered with the name Creature. (Some survived and therefore bore the name all their lives.)
In the 17th C Midwives also had a duty to bury unbaptised and stillborn babies secretly in a hidden place.https://oxfordindex.oup.com/view/10.1093/oi/authority.20110803110604183