Let us suppose that the Judaeo-Christian deity is an aspect of the seasonal deity of the summer season, known as the Summer King (or perhaps the other way around, the Summer King being an aspect of the Judaeo-Christian god, it doesn't really matter). The Summer King has - in a largely but not entirely disconnected set of magical realms - become increasingly dominant over the deities of the other three seasons in the last few thousand years, particularly in the last thousand or less, so that the Spring Princess and the Autumn Prince deities' avatars have vanished entirely, and the Winter Queen has been greatly weakened.
The end result of this is that the worlds - including our world - are all headed toward a catastrophic global warming disaster.
The Winter Queen has established a plan to restore balance to the seasons (and to bring the magical realms closer to our technological realm), not to seek total dominance as the Summer King has done.
Part of this plan is for the Winter Queen to (re-)establish her religion in both the magical realms and our technological realm.
How might the Winter Queen go about doing this given:
The deities can create human (to some degree super-human) avatars, but doing so is expensive in terms of magical power. The Summer King has one, and could conceivably afford only a handful more. The Winter Queen has one, and cannot afford more yet. The Spring and Autumn deities cannot afford even one, and will require the assistance of their temporally-earlier deity to create one.
The deities promote particular personality traits and lifestyles in their worshippers as well as favouring their particular season. The Summer King favours gender inequality, slavery and servitude, chastity or celibacy (just look at the Old Testament), summer being a season of physical warmth but emotional coldness.
The Winter Queen, being in opposition, favours gender equality, emancipation and sexual freedom, winter being a season of physical coldness but emotional warmth. As a specific example, since the Summer King vilifies prostitutes and supports them only if they give up their profession, the Winter Queen supports them in their profession.
The Summer King is powerful (due to lots of pre-existing worshippers) but is lazy and complacent, and hoards his magical power rather than sharing it with his worshippers (hence few if any real miracles occur and few if any divine powers are handed out to the more devoted worshippers). If any powers were to be handed out, they would be in line with the powers and abilities described in the Old and New Testaments, but they would be exceedingly rare. Assume that this won't happen for a good many years.
The Winter Queen is much less powerful, but can potentially offer especially devoted worshippers a number of advantages, including immunity to exposure to winter conditions, better performance in winter sports and activities, immunity to sexually-transmitted diseases, and for women, the ability to start or stop their menstrual cycles at any point they want. A very few particularly devoted worshippers of the Winter Queen may be able to read the minds of others at ahem "touch" range, and share what they learn with their goddess. All this comes at a price, in that the Winter Goddess must obtain more worshippers than the Summer King (on the order of, say 25% more) to attain the same level of personal power. Of course, if the Summer King was to begin performing miracles and handing out powers, the relative worshippers to power ratios would go back to parity.
The Winter Queen must eventually assist the Spring Princess and Autumn Prince to regain their lost powers and to create new avatars. As the Winter Queen's avatar must literally give birth to the Spring Princess, that is relatively easy to achieve once Winter becomes a bit more powerful, but the Summer King's avatar must father the Autumn Prince's avatar, and must be either convinced or tricked into doing so.
Neither the Spring Princess or the Autumn Prince can assist in the timeframe of this question.
Our technological realm is currently pretty much ignorant of real magic, but this can - and indeed must - change as the Winter Queen acts.
Since it would take too long here to describe the many magical realms - suffice to say that each is roughly the size of a kingdom, big or small, and without magic, travelling in one direction just gets you back to where you started from pretty soon - answers should concentrate on how the Winter Queen might go about (re-)establishing her religion in our current-day magically-ignorant technological realm.
The timeline I mentioned in a comment of 10-15 years is for a reasonable start, not parity. The Winter Queen really only needs a relatively small share of the world's religious devotion over that period, perhaps 5-15%, and ultimately needs around 25%, which may be shared with other deities, as the Winter Queen, unlike the Summer King, is not a jealous deity.
Directly co-opting or subverting the Judaeo-Christian religions (Judaism, Christianity, Islam) is out of the question for multiple reasons:
The Winter Queen is the Antichrist and the Whore of Babylon as far as these religions are concerned, and her theology would be quite antithetical to that of these religions.
The Summer King is lazy and complacent, and might not notice a gradual trickle of worshippers away from him, but if the Winter Queen started a wholesale takeover, that would get him interested pretty quickly, at which point there would likely be a fight that the Winter Queen - and possibly the rest of the world - would lose.
Part of the Winter Queen's strategy is to present the Summer King with a situation that he cannot win, without having to make him lose either. This is about balance, not dominance. If the Winter Queen was to actually win, there would be an ice-age, and while she wouldn't really mind on a personal level, it wouldn't be good for her worshippers, and unlike the Summer King at present, she cares about them.
Deities can hand out powers to perhaps 5 to 15% of their worshippers, as long as that 5-15% results in a whole worshipper - no fractions allowed, so it would take at least 7 worshippers before one could gain a power. If the worshippers with powers tend to have more powers each, then fewer in total can have powers. The worshippers who can gain powers must be highly devoted, have compatible personality traits, and must also be proactive rather than passive types of people. Typically around 50% of those will gain one power at most, 25% will be able to gain up to 2 powers, 12.5% will be able to get up to three, and so-on, individuals with one more power than others will be half as common, though this may vary somewhat from deity to deity - some may concentrate their powers in fewer individuals, others may spread them out more. This comes at a cost of around 25% of the deity's personal power, so by not handing out any powers, the deity gains a personal advantage.