Section 5.) Although it be a great sin to contemn or neglect this ordinance,(1) yet grace and salvation are not so inseparably annexed unto it, as that no person can be regenerated or saved, without it;(2) or, that all that are baptized are undoubtedly regenerated.(3)
(1) Lk 7:30 with Ex 4:24-26 (2) Ro 4:11; Ac 10:2,4,22,31,45,47 (3) Ac 8:13,23
Section 6.) The efficacy of Baptism is not tied to that moment of time wherein it is administered;(1) yet, not withstanding, by the right use of this ordinance, the grace promised is not only offered, but really exhibited, and conferred, by the Holy Ghost, to such (whether of age or infants) as that grace belongeth unto, according to the counsel of God's own will, in His appointed time.(2)
(1) Jn 3:5,8 (2) Gal 3:27; Tit 3:5; Eph 5:25,26; Ac 2:38,41
Westminster Confession Chapter 28 Of Baptism
According to the above statements of Section 5 and 6 - especially the parts I have emboldened - the other statement (as quoted in the question from the Book of Common Prayer) is in accord.
However the texts listed are not expounded upon, they are just mentioned without explanation. Nor could I find any more full discussion regarding the question of the transition of a child from the stated state of 'regeneration' to an alternative state by the act of committing a sin.
I had hoped for more detail on this from the Shorter Catechism but I have not been able to find any, only the bare statement of :
Q: What is baptism? A: Baptism is a sacrament, wherein the washing with water, in the name of the Father, and of the Son, and of the Holy Ghost, (1) doth signify and seal our ingrafting into Christ, (2) and partaking of the benefits of the covenant of grace, (3) and our engagement to be the Lord's. (4)
Again, the texts are not expounded, just listed, so there are many questions (in my own mind) regarding the logic of what is being presented by the three Anglican sources (viz. The Book of Common Prayer, The Shorter Catechism and the Westminster Confession) as an explanation of the content of The Holy Bible.
As the question indicates, I suspect a more full explanation is only forthcoming from individuals within Anglicanism (as stated, 'Cranmer' or other 'Reformers') but that will only express their own position and will not be, per se, the overall stance of the Church of England.
EDIT AFTER COMMENT
With thanks to the OP for comment, I also looked at the Thirty Nine Articles and found no precise explanation. :
BAPTISM is not only a sign of profession, and mark of difference, whereby Christian men are discerned from others that be not christened, but it is also a sign of Regeneration or new Birth, whereby, as by an instrument, they that receive Baptism rightly are grafted into the Church; the promises of forgiveness of sin, and of our adoption to be the sons of God by the Holy Ghost, are visibly signed and sealed; Faith is confirmed, and Grace increased by virtue of prayer unto God. The Baptism of young Children is in any wise to be retained in the Church, as most agreeable with the institution of Christ.
The Thirty Nine Articles - XXVII Of Baptism
The wording stresses 'grafting into the church' and that 'baptising children (sic)' is 'agreeable to the institution of Christ' but does not expand on what 'institution' might mean.
Also baptism is called a 'sign' of regeneration but is not overtly stated to be regeneration in and of itself.
So it still appears to me that there is a distinct lack of substantial explanation of the Anglican doctrine of baptism, despite sufficient documentation regarding the actual practice of it.