Do paedobaptists suggest that a baptised infant is really 'born again'? This question is for Protestant Trinitarians and addresses the divide among the various denominations of Protestantism, particularly Presbyterianism, Congregationalism and Anglicanism, who baptise infants rather than adults.
I was, myself, 'baptised' as an infant, but late in infancy as my father was a Church of Scotland clergyman in the remote Highlands of Scotland and was not permitted to baptise his own child, so the matter was delayed and I remember it vividly at the age of five. But after conversion at the age of sixteen I volunteered to be baptised in a Baptist assembly (and my father graciously attended).
Among all the arguments to and fro about circumcision and 'sanctification' it is still not clear to me if those who baptise infants actually claim that the child is 'born again' in the various senses in which that term is used in the New Testament, that is to say : born from above (anothen); born anew (anagennao) ; born again (paligenesis) ; born of God ; and born of water and Spirit.
What exactly do paedobaptists believe is happening (in New Testament terms) during the ritual of placing water on a baby ?
And if they regard it so (that this is genuine 'new birth') then can such infants not be recovered if they (initially) reject the gospel in childhood since that would be regarded (in New Testament terms) as being a castaway, having been 'born again' only to reject Christ ?
You will probably find different denominations have a slightly different flavour to this answer. To put my cards on the table, I come at this from an Anglican perspective.
My first point would be, what do credobaptists think is going on in communion and how is that different to paedobaptists? Most of the credobaptists I know, if I understand correctly, do not think that baptism washes away sin - that event has happened with conversion; baptism is just the sign. It's the same with paedobaptists as well. I don't think that anyone from protestant churches would actually say that baptism and regeneration were one and the same event.
I know of people who were baptised as teenagers, then went away from faith, then came back to it later in life. I can't see how this is really different from being baptised as a child and then going away from faith before coming back to it.
According to Lee Gatiss' booklet &quot;From Life's First Cry: John Owen on infant baptism and infant salvation&quot;, infant baptism is not so much about affirming faith but rather &quot;the promise and covenant of God&quot; (p. 24). John Owen argues from the fact that, since children of believers are part of God's covenant, they should be considered within God's covenant people and therefore have a right to the sign of baptism.
So he says:

This is why impenitent sinners are not to be baptised, because not having the reality they have no right to the sign. But he goes on to say that if the sign is denied to believers' infants then that implies (by the same logic) that God is denying them the reality of salvation too - &quot;and then all the children of believing parents dying in infancy must, without hope, be eternally damned.&quot; ... It is certainly  not that for Owen baptism is essential for salvation, but rather that if God desires the sign of salvation to be withheld it must be because he withholds salvation itself from such infants. [My emphasis] (p. 30)

I would say, paedobaptists baptise infants who are too young to profess faith because they belong within the covenant people of God. This isn't tying in baptism with regeneration, but rather - by virtue of belonging they have a right to the sign of belonging.
So my answer to your final question is - yes, infants baptised who then go away from Christ can be recovered and do not need re-baptism. (In the Church of England we have an order &quot;re-affirmation of baptismal vows&quot; which is often used for occasions when someone comes back to faith but is already baptised). In fact I would suggest that to go away from Christ in New Testament terms is an eternal thing: someone can go away but then come back again. The final moment of being cast away and rejecting Christ doesn't happen until death.
Additionally: I saw this Twitter thread yesterday and I think you might find it helpful. He quotes Herman Witsius:

God has given that pledge to pious parents that they may regard their little ones as the children of God by gracious adoption, until, when further advanced, they betray themselves by indications to the contrary and that they may feel not less secure regarding their children dying in infancy than did Abraham and Isaac of old.