Given the kinds of websites that are publishing this poem, I don't think it's a stretch to say that this change has something to do with race or racism. And the thing about race and racism is that these ideas are inseparable from gender and reproduction.
There's a reason why so many states and countries worked very hard to pass anti-miscegenation laws--laws that segregated marriage. There's a reason why the justification given for lynching black men in the American south was that they had allegedly raped white women. Race is seen as something passed on through birth; according to racists, in order to maintain the white population, it's important for white men and women to marry and have white children. (Never mind that this is contradicted by modern genetics; ideology trumps science any day of the week). So if a white woman--who is seen as the rightful property of a white man--has a relationship with someone who is not white, to a racist that is problematic because they are allegedly causing the "white race" to decrease in population and die out.
This is certainly something that concerns the people who are publishing the poem. For example, the website europeanamericansunited.org has, next to pages titled "Diagramming a White America" and "Diversity is Our Weakness", a page called "Ethics and Mate Selection." The page is full of alleged scientific research about how humans "select mates" and advice for "selecting mates." The subtext--that "selecting a mate" is important to preserve the "white race" isn't stated explicitly in that particular page, but you don't have to look hard to find it:
Yes, the possibilities are endless; but those possibilities can only exist so long as the forces of Dissolution and Chaos do not succeed in their evil and genocidal plan to amalgamate the wonderful diversity of humanity into a medium-tan "United Nations Man" in their rapacious lust for power and all-consuming greed to unjustly expropriate the labor of others. They want an alienated unit of production and consumption with no history, no culture, and no future beyond created consumerist desires. A creature with no ancestry that can serve as a source of pride or a well of patriotism that could lead him to stand against his evil overlords. The medium-tan United Nations Man is NOT forward progress in the evolution of Life, but rather the surrender to entropy in the devolution to inert and homogenized matter. It is the opposite of life: it is DEATH.
Genocide is evil -- uniquely evil -- because of all the crimes that can possibly be committed, it is the ONLY crime that carries with it the very real potential of ending ALL human life. Genocide can take place through direct killing, to be sure, but can also take place through intermarriage. The disappearance of an entire race of humanity, whether a little at a time or all at once, is still the same thing. And the extinction of even ONE race of humanity could very likely mean the ultimate extinction of ALL of humanity. Chew on that for a minute.
(Taken from the page titled "The EAU View of Race"; I'm trying to avoid linking to these pages for obvious reasons, but you don't have to look hard to find the pages I'm quoting from).
To move on (finally) to a discussion of the actual poem, I think the best way to interpret the change from "the English" to "the Saxon" is that Saxon is a reference to the lineage of the "white race". Of the four words "White Anglo Saxon Protestant" (WASP is an acronym that stands for a particular group of white Americans), Saxon is the word that sounds the most like a historic group of people far off into the past. Referencing this far off ancestral group is referencing the idea of an unbroken chain of reproduction, a chain that must be continued at all cost. (As I've established in the previous paragraphs, this is something that the people publishing the poem find to be crucially important). It's not unlike a King or a Ruler basing their claim to power on the fact that they are descended from previous Kings or Rulers. The reference to the Saxons establishes whiteness' legitimacy--the "white race" has been around for a long time--and also references a need for whiteness to continue reproduce in the future.