First of all, as you note not all of these names really ARE of Jewish origin. Peter, Mark, and Andrew, for example, are of Greek/Roman origin. Thomas... it may technically have Aramaic origins but it is definitely not the descendant of any Hebrew name still currently in use. Explaining the others requires more discussion of Jewish naming practices (let's confine it to currently-English-speaking countries given that you framed your question about Anglicization). For the purposes of this question, I'll be giving all Hebrew names as the Hebrew versions.

For centuries, Jewish men and women were named in different ways. Broadly speaking, it was only important for men to be given a Jewish name, as the name by which they were given at their circumcision and called up with to the Torah. But, of course, often people will want a name that more easily blends in with their surroundings- and that's what led to the creation of the kinnui, essentially a nickname. There were lots of variants on this- it could be a whole name, like how Rabbi Moshe ben Nachman, or Ramban/Nahmanides, was also known by the Catalan name Bonastruc ca Porta, in his work as a doctor in Girona. A kinnui could be a translation of a name, so that in Arabic lands, Ovadia, for example, could become Abdullah (they both have the same meaning, "servant of God"). There were comparable ways of creating kinnuim- whether just a secular name that they liked or an actual translation of their name/name of comparable sound. (There was even a Rabbi Peter in medieval France! That said, your instinct is correct- it was an EXTREMELY uncommon name.)

In European lands, Ashkenazic Yiddish-speaking Jews developed many such kinnuim to the point that Jews began to be named/known by BOTH names, as a double name- a famous example would be Maharal, Rabbi Yehuda Loeb/Loew ben (son of) Bezalel, whose first name is his Hebrew name (which Anglicized would be Judah) and whose middle name is a Yiddishized/Germanized version- not a TRANSLATION of Yehuda, but of the Hebrew word (and name) Aryeh, which means "lion." The reason for this kinnui is that in the blessings of Jacob to his sons at the end of Genesis, Yehuda is compared to a lion- so it was seen as a suitable "translation" for the name. There are many similar kinnuim- some which seem somewhat intuitive, like Dov/Ber (which both mean "bear") and Baruch/Bendet (which both mean "blessed"), and some of which are less so.

Of course, the most common way that kinnuim came about was just to find a name that was similar to the Hebrew name- Yiddish kinnuim were no different in that regard. So someone named Shlomo might have as his kinnui Salomon, the German version of the name- which then turned into the Yiddish name/kinnui Zalman. The name Asher acquired the kinnui Anshel, etc. As time went on and Jews became more acculturated, they began to pick names that were more explicitly secular so that they could blend in. So Moshe, which COULD have been Moses (the version in the translated Bible), instead became Moritz in Germany, Maurice in France, Morris in the US- much less old-fashioned than Moses! Of course, some names just used the translated Biblical versions- like Avraham becoming Abraham and Rivka becoming Rebecca (or whatever the local equivalent would be).

To turn to women- in both Ashkenazic and Sefardic lands, women were much more likely to only be named a secular, or local, name than men were. A woman could be Esther or Sarah, but she could also be Fortuna or Allegra in a Spanish/Ladino speaking community, or Jamila or Aziza in an Arabic speaking community, or Beila or Perel in a Yiddish speaking community. In some cases, there was something of a kinnui component to it, such as Chana becoming Gracia in Spanish or Henna in Yiddish, but the concept of kinnui was not considered as necessary in and of itself because women didn't need the Hebrew names in the first place.

Moving toward modern times, in English speaking countries (and, to be honest, in Europe as well), the goal of having a secular name (generally in addition to a kinnui!) was to be able to use it in greater society. So naturally, in cases where the Hebrew name (or, as time went on, the Yiddish name/kinnui which had acquired the force of Hebrew name) was given first, a secular name would be found to follow that would fit in with the local fashion. It didn't necessarily have to have any real relation to the Hebrew/Yiddish name- Shaindel might become Florence, for example, just because Florence was a popular name at that time. To the extent that any of the names you mentioned above are popular among Anglo Jews, Elizabeth is probably the one that is most popular- and it's not because it is a variant on Elisheva, which is a bit of trivia that the average Jew likely doesn't even know. It's because the name itself was popular in greater society, and an Elizabeth could really have any Hebrew, or Yiddish, name, whether Elisheva or Esther or Yehudis or Breindel.

With regard to the rest of the names on your list- starting with Nathaniel, that one is only less popular because the name which it is an Anglicized version of, Natanel, has had phases of popularity, like any other name. Not everyone who uses the Hebrew name will give Nathaniel as the English name, but the Hebrew name is definitely in use. For the others- all of them HAVE been used, but as you note the English versions just never became popular. Part of it is because many of those names are only translations of Hebrew names IN THE NEW TESTAMENT. James is technically based on Yaakov, but in the translations of the Hebrew Bible that's actually Jacob, which is more popular for Jews to use. Likewise with Miriam, which is technically the origin of Mary but who is still called Miriam in English translations of the Hebrew Bible/Old Testament. Yochanan and Matityahu are just not particularly popular Hebrew names, and since they're not Biblical (at least not for Jews) there would be less of an instinct anyway to use John/Matthew as a translation- there's no immediate association between the names among Jews. A boy called Yochanan would be just as likely to be called, say, Jeffrey as John.

Now, if any of those names could be used as a secular name by Jews, why aren't they? After all, we just established that James doesn't have to be Yaakov but could just as easily be Yom Tov Lippmann. (For a real life example of this, the formerly-chassidic reggae artist Matisyahu was born Matthew Miller and when he became religious initially, he named himself Matisyahu because that was the origin of the name Matthew- only for him to discover that at his circumcision he'd been named Feivish Hershel, a VERY Yiddish and old-fashioned name that I can't blame him for jettisoning.) The answer is that some of them are, like the aforementioned Matthew Miller, Rabbi Peter, and every Jewish woman you may ever meet named Liz or Beth. But they will never be specifically popular names BECAUSE of their Jewish origins, and yes, a name like Mary or Peter might have a bit more overtly Christian an overtone than Rebecca or Jacob and thus be less appealing.

In general, though, Anglicized versions of Biblical names will follow popularity cycles, like any other name. Sometimes they will be based on the popularity of the HEBREW name- in Orthodox Jewish communities, for example, people are more likely to go by their Hebrew names in daily life, so Esther, while a decidedly old-fashioned English name, is a perennially popular Hebrew name and so will be given as a legal name, while Manoach or Tzefania or Choglah never quite caught on and so neither did their English versions. Sometimes, though, they'll be based on the popularity of the ENGLISH name- a family might choose to name a son Avraham, decide that Abraham is too archaic, and give Avraham the English name Alan or Andrew or Avery instead. And sometimes, fascinatingly, an Anglicized Biblical name will be used instead of a given Yiddish name- almost as a reverse kinnui- as in someone I know whose Hebrew/Yiddish name is Roiza but goes by Rebecca.