A denizen is an alien born, but who has obtained ex donatione regis letters patent to make him an English subject: a high and incommunicable branch of the royal prerogative. A denizen is in a kind of middle state between an alien, and natural-born subject, and partakes of both of them. He may take lands by purchase or devise, which an alien may not; but cannot take by inheritance: for his parent, through whom he must claim, being an alien had no inheritable blood, and therefore could convey none to the son. And, upon a like defect of hereditary blood, the issue of a denizen, born before denization, cannot inherit to him; but his issue born after, may. A denizen is not excused from paying the alien's duty, and some other mercantile burthens. And no denizen can be of the privy council, or either house of parliament, or have any office of trust, civil or military, or be capable of any grant from the crown.
It appears to me, that the only aliens whose issue can be "natural-born citizens" are those who formally swear allegiance or "actual obedience" to the crown ... such as an alien who has applied for denizenship. The child of a "denizen" born BEFORE denization cannot inherit (i.e., is NOT a natural-born subject), but the one born AFTER denization CAN inherit and is thus a natural-born subject.
This is the same in the U.S. basically as not being a citizen at birth until AFTER the father naturalizes.
What the Blackstone passage doesn't make clear is that a "denizen" BEFORE denization is just an alien, therefore the child of such a person cannot inherit and is therefore NOT a natural-born subject. Once the alien becomes a denizen with actual obedience and allegiance to the crown, THEN the child may inherit and is thus a natural-born subject. This seems to be missed when the interpreters of common law say they children of aliens are natural-born subjects ... this is only for the children of aliens born in the country AFTER the denization of the father.
That is a very good point, and that is also what I seemed to be reading in Bacon's book cited above. The fact that such people are treated differently in Inheritance law demonstrates pretty conclusively that the English Common law did not really regard them as the same.
While poking around a bit more, I found this interesting 1915 essay by F. T. Piggott explaining why he believes English Common law is such a mess regarding British Subject status. He echoes a concern mentioned by Patlin above. (That of forceable Patriation.) Not that any of us needs something more to read! :)