The main finding seems to be that, for whatever reason, most Greek god names were not in particularly wide usage in English before the mid-nineteenth century, when many of them saw a notable uptick in frequency that has continued through to the present day.
My approach has been to stick various god names into Ngrams (generally with the timescale 1800–2008; 2008 being the most recent one can go) and see what comes out. The big caveat here is that we can’t always be sure if the words are actually being used to refer to the gods themselves: often they might be applied to other things. For example, Mercury, Venus, Mars etc. are the names of planets; Hermes is a popular name for postal services; Hades is sometimes used to mean “Hell”, etc.
Sometimes the Greek form has been around for quite a while. This is the case with Hermes for example:
You can see a not dissimilar graph for yourself if you compare Hades and Pluto. Note, though, that the Roman form is still generally the more frequent.
In many other cases, though, the Greek form only really starts to be used towards the middle of the 1800s, with a gradual overall increase in frequency since. This is the case with Poseidon, for example, which only really started to be used in about 1830, with another big jump in the 1860s:
There may be a correspondence between increased use of Greek Poseidon and decreased use of Roman Neptune, suggesting the first is being used in place of the second.
We see a similar pattern with Greek Zeus and Roman Jupiter/Jove – here, the increase in the Greek form at the expense of the Roman across the nineteenth century is quite marked:
The Roman forms remain more popular than the Greek, however.
Greek Aphrodite and Artemis behave similarly (opposed to Roman Venus and Diana), though the Greek forms in these cases has never been very popular. Again, you can see this if you put the names into Ngrams yourself. Greek Dionysus has gone from being very rare to competing with Roman Bacchus in frequency:
Greek Eros shows a similar pattern as opposed to Roman Cupid. Greek Persephone, barely used 200 years ago, is now maybe three times more frequent than the Roman Proserpine.
The Greek Hera, Athena (= Roman Juno, Minerva) were less relatively unfrequent to begin with, but there’s still a notable jump at a similar point in time, and nowadays the Greek and Roman forms are about as common as one another, as the graph for Athena and Minerva shows:
Some Greek forms have never been popular. This is the case for example with Cronos (poor Cronos):
Ares (=Mars) and Hephaistos (=Vulcan) have similarly remained unpopular. However, with all three if we leave the Roman forms off the graph we can still see an increase in frequency in a similar time period to before:
The general pattern then, to reiterate, is that many Greek forms only really take off in English in the mid-1800s, though there are a few exceptions. In some cases, the Roman forms remain more popular; in others both variants are now about equally frequent. In at least one, it’s the Greek form that is now dominant.
I don’t know what exactly the explanation for these patterns is, but perhaps someone else does.