To take the best-known passage alone, rather than the whole speech, and skip over duplicates, we can quickly show that this is the case:
we Old English we
shall Old English sceal
fight Old English feohtan
on Old English on a variant of Old English an
the Old English þe, from earlier Old English se
beaches Old English bæce/bece
landing Modern English noun from land, from Old English land/lond
grounds Old English grund
in Old English in
fields Old English feld
and Old English and/ond (in which it at first only had the "then","next" meaning rather than the "and also" meaning).
streets Old English stret in some dialects, stræt in others.
hills Old English hyll
never Old English næfre
surrender Old French surrendre
Now, the next question is to what extent does it tell us something about English, and to what extent something of Churchill's mastery of it?
The speech is in general not one that attempts to demonstrate Churchill's intellectual mastery of a subject (to which we turn more often to words of Latin and Greek origin), or which requires technical precision (in which we turn to the terms with most precision in a field, which tend to be Latin or Greek again), or his florid emotions (to which we turn more often to the Romance languages, which in turn owe a lot to Greek). No, he wanted to underline a strong, almost crude drive, simply expressed.
The speech called for simple words, and the simplest words in English - those that are known to everyone from a young age - are mostly Old English in origin. Old English is the core of English after all (that's why it's called Old English). Even a speech that made heavy use of Latinate words, would have much that came from Anglo-Saxon (all the articles and pronouns for a start).
Even if he was aware he was using Old English words heavily (and as the historian whose works included the four-volume A History of the English-Speaking Peoples, it's quite likely that he would have been conscious of it), his primary concern was more likely this simplicity. He was not going to make an intellectual argument in favour of continuing the war, he was going to drill in a series of similar emotive assertions that would encourage the hearts of his fellow politicians, his fighting men, those otherwise involved in the war-effort, and the general populace, alike.
And most of us would turn to mostly Anglo-Saxon words in such a case too.
He may have been considering the emphasis upon Englishness suggested by this. He would probably not have wanted to do this too blatantly. After all, the allies where not solely English (in the context of the speech, they were mainly French), and even those in British uniform included many from elsewhere in the UK and its colonies, along with some volunteers from elsewhere. I think, too much is made of this possibility.
So too, do I think too much is made of the word surrender being of Old French origin. In recent years much has been made of this, particularly in response to some in the US attempting to stereotype the French as militarily weak after the French government opposed the invasion of Iraq. Aside from ignoring just about everything about French and US military history, it's doubly irrelevant here.
For a start, the speech was made during the Battle of France. The BEF were fighting along-side an International force that was mainly French, to keep France free from the German invasion. This is not a time where Churchill would have insulted the French!
Second, we can consider the effect of this word by changing "We shall never surrender" to "We shall never give up", which would make the passage entirely Old English in origin, but the words "give up" differ from "surrender" in both nuance and euphony. In addition to being almost all Old English, the words are almost all short, being mostly monosyllabic with just two two-syllable words (one of which is part of a three-syllable noun-phase). The three-syllable surrender allows a slightly longer consideration of that assertion — even though it's a negative one — of what they would not do, coming as it does after his previous positive assertions of what they would do.
In all, I think the heavy use of Old English says more about English than it does about Churchill's use of it. This is not to deny thatChurchill was a master of the language, and his speech was a great example of that mastery. Many other eloquent speakers and writers of English would similarly have used Old English heavily if they wanted to evoke a similar response in their audience.