Which native English speakers are linguistically the most "germanic"?
Solution 1:
There's no really good answer to this question, but we can take a stab at it if we accept some very broad generalisations.
English is a Germanic language by virtue of being descended from Proto-Germanic (which is a matter of geography and historical migration patterns). Setting aside the question of English dialects for a moment, among all the Germanic languages we can say that a language is "more Germanic" if it has undergone fewer changes since splitting from Proto-Germanic. The ideal way to do this would be to do a detailed inventory of linguistic changes apparent in a language and analyse how many steps might have been taken to get from Proto-Germanic to the modern language, but that's a load of work. A very broad generalisation that is less work would be to look at how many forks in the family tree exist between a language and Proto-German. This is much easier to see. (As an amusing aside, you can see from the family tree that Swedish is—among several others—"more Germanic" than modern German.)
By analogy, we can say that the English dialect that is closest to Proto-Germanic is the English dialect that is closest to an earlier branch in the family tree. Or, put another way, the oldest dialect of English will be the closest to Proto-Germanic, and hence the "most Germanic" dialect of English.
We can quickly rule out any English dialect outside of the British Isles, since English dialects in the rest of the world are descended from some form of British Isles English, and hence would be at least one more step removed.
Of course, such a detailed analysis of English dialects is also prohibitively complicated, but at least it gives us the conceptual framework to figure out that "oldest British Isles English dialect" is what we're looking for. So, we'll have to go with anecdote and hearsay: the Geordie dialect is the oldest dialect in the British Isles, according to current opinion.
Solution 2:
The answer is all of them. English is a Germanic language.
A far better question to ask would be "What other Germanic language is linguistically most like English?" (Hint, the answer would not be "German").
The problem with the original question is that it seems to imply that the language we today call "German" is the root of the English language, and thus there must exist some dialect of it which has diverged the least from "true German".
What instead happened was that some speakers of a (low) German language moved to England and slowly their language developed on its own, mostly isolated from the original tongue on the continent, into what we call "English". Meanwhile people on the continent had their own dialects which evolved into what we today call separate languages such as "German", "Dutch", "Danish", "Frisian", "Norwegian", etc.
All of those, and English too, are "Germanic" languages. Whatever exact language was spoken by the early German immigrants to England (let's call it "Germanic"), modern Germans would be no more able to understand it than they can English. If you don't believe me, pick up a copy of Beowulf (written in Old English) and see how easy it is to read.
Oh, and I understand the answer to the question I posed, is "Frisian". It is the sort of light-tan area in the map of Germanic language areas below.