This is going to strongly affect public attitudes toward space travel, the existence of aliens, and the paranormal, among other things. The arrival of the probe from Dp-Earth, even more so, and in ways that will not be obvious, at first.
When the last booster burns up in OTL-Earth's atmosphere, either the thoughtforms within the booster's unspent fuel, or the entity that caused all this to happen in the first place, give OTL-Earth a 'gift': somewhat less than 1% of the population, all over the world (though not consitent with population density - North America, Europe, Japan, and
Oceania (mostly in Oz and NZ) tend to have more psykers than either India or China, though India has more than China does) develop active psi powers - mostly low level, and mostly just thoughtform generation. The selection of exactly who gets powers is somewhat telling, as all of them tend to be considered strange within their own nations and cultures: Wiccans and other neopagans, New Age mystics, particularly 'spiritual' martial artists, tribal shamans (they're respected, but do tend to be seen as 'a bit off'), followers of Vodoun and Santerķa, science fiction and fantasy fans, gamers, the more eccentric historical reinactors, would-be
superheroes (well, the ones that are relatively serious about it), and mentally ill folks (OK, there's
some overlap between the last two, particularly if chuunibyou is a real mental illness), in or out of asylums, with sufficiently 'interesting' derangements. Few of these people start with even enough power to hover, much less fly, nor are they going to be bouncing bullets off their chests any time soon. Those who get psi powers beyond thoughtform generation and manipulation tend to start with subtle abilities like probability alteration, ESP, and sometimes astral projection, meta-psi, psychic vampirism, and telepathy, rather than anything flashy, and even those with more obvious powers like PK or EK are rarely going to have something impressive - expect one to levitate a pencil, not oneself. Also, mind control is far from easy, more along the lines of telepathically-assisted RL hypnosis than the sort of things you see in the comics.
The abilities gained are generally consistent with the individual's beliefs, interests, and personality - someone who believes they have a particular power, or strongly wants it, will gain some version of it, albeit generally an unimpressive version, at first - Star Wars fans who 'feel the Force', martial artists with (weaker versions of) the legendary and/or cinematic powers of their style, D&D gamers with a Path style based on the Schools of Magic in that game, or a Book style based on the feats and powers of their favorite class (at barely first level), and so forth. Strong-willed individuals, as well as those who don't take orders well (which isn't quite the same thing), tend to start out stronger, and improve faster, as do those with greater intelligence (though a combination of all three is best). Those not given the gift can train for it, particularly if taught by someone who already has it (though again, the ones who don't take orders well tend to be better at it) - this is rather easier if the teacher is a telepath, or has learned or developed mind-affecting spells.
Some animals will also develop powers - mostly cats, and species native to Australia, though any animal of a species with significant local myths about it might merge with an appropriate spirit (e.g. foxes and tanuki in Japan, coyotes and ravens in North America, and so forth). This rarely happens to plants or locations, without at least semi-conscious human interference - at least until something arrives from Fa-Earth.
Being so affected by Dp-Earth, the spirits tend to behave 'in-genre' (though they don't all agree on what genre it is), and try to get machinery, animals, and people to behave in-genre, as well - this doesn't work well people, especially as in-genre behavior is often the opposite of what a professional is trained to do (listen to a cop or soldier complain about cops or soldiers in most fiction, and you'll see what I mean). It rarely works well on animals, unless it's consistent with their instincts. This in-genre behavior may be a factor in who gains powers, and who is better at it.
It isn't going to take long before governments start asking their space agencies to draw up plans for both manned and unmanned visits to other worlds, and getting frustrating results. Then again, for an interplanetary spacecraft, the
Nautilus-X looks surprisingly affordable, even if you add a second grav-wheel to neutralize the centrifugal force - how affordable it turns out to be is another matter, but given the gift of magic, 'less expensive' is actually fairly likely. Before that, though, the difficulty of space launches from OTL-Earth will be at RL levels - complex and very expensive. OTOH, the situation on Dp-Earth will likely lead to various nations' military budgets being diverted to space, even before the probe arrives.
On another note, the particles produced by dimensional transition have a sufficiently consistent speed, that observation of particle detectors can precisely locate any dimensional incursions heavier than 3 kilograms, even if astronomers somehow don't (which isn't so helpful with spirits, but while they lack mass, they do disturb whatever it is that's producing these sprays of particles as if they did - just not much).
Per samdamandias' suggestion on SB.com, these worlds are designated Crossroads-4a through Crossroads-4e. For purposes of the IW setting, Stp-Earth is on Quantum 3, Clp-Earth is on Q4, OTL-Earth is on Q5 with Homeline, Fa-Earth is on Q6, and Dp-Earth is on Q7, unless someone with more knowledge of the IW setting gives me a good enough reason otherwise.
Thinking about it, I feel sorry for any Infinity agents that show up on OTL-Earth: Their arrival point is likely to be spotted immediately (due to the particle spray mentioned in the OP), if they arrive after 'we' start triangulating such bursts (something I can see happening fairly quickly), which is bad enough, but imagine having to give your boss a copy of GURPS Infinite Worlds. Sure, they probably won't find that first thing, but they ought to stumble across it eventually, if they don't just never come to this world. If they only arrive on one of the other Earths, though, they'll be able to stay hidden for rather longer, because you need 21st century tech to detect the particles (I think; I'm not up in the latest in particle detection tech, nor how old it is), and it'll be a while before we set up a detection scheme for the other worlds.
Timeline (OTL-Earth dates used, for convenience):
2012 (December 21): The new Earths appear, the nearer ones first, and the further ones a little less than thirteen seconds later. From the prespective of the other Earths, of course, the order of appearance is somewhat different. Communication attempts will begin in short order, once the right frequencies and signal strengths are found, but only Dp-Earth and Stp-Earth are equipped to respond. At some point on the 21st or 22nd, Winston Churchill has a frustrating conversation with David Cameron (sadly, it would be at least as frustrating with a PM of any other party), and begins to wonder if he'll get more help, faster, from the folks sending radio-telegrams; he is heartened, though, by Merkel telling Hitler where to shove himself and his Reich (even if hardly anyone on Dp-Earth thinks a woman is
really Chancellor of Germany).
Dec 22: An astronomer on OTL-Earth notes an asteroid near Clp-Earth, moving away, and marks it for further observation.
December 26: A probe is launched from Dp-Earth, and begins to be mated to the booster rockets already in orbit.
Dec. 28: The probe leaves Dp-Earth orbit, on course for OTL-Earth.
Dec. 31: The asteroid observed near Clp-Earth disappears at 3 light-seconds from that Earth.
2013 (January 4): The Dp-Earth probe fails to disappear.
Jan. 12: the probe from Dp-Earth passes through an asteroid, unperturbed. The asteroid is only visible from OTL-Earth, not the others.
Jan. 19: The probe enters the three light-second bubble around Earth, releasing a significant particle spray.
Jan. 21: On OTL-Earth, the UK requests an emergency summit of the leaders of all members of the Commonwealth. Rumours surface of sightings of soldiers guarding a blue police box in the woods of Northern England, and of a man bearing a suspicious resemblance to Jon Pertwee being taken to a cottage hospital nearby. The placing and quickly rescinding of a D-notice on these matters only encourages speculation (which may have been the point).
Jan. 26: The probe arrives in OTL-Earth orbit, its last booster falling into our atmosphere, and burning up. People on OTL-Earth start developing (generally minor) powers.
February ??: The members of the Commonwealth of Nations begin ratifying the Treaty of New Delhi, the founding document of the Unified Intelligence Taskforce, or 'UNIT', a military group dedicated to the defense of the Commonwealth and it's members and peoples from threats of an extraterresial, paranormal, or otherwise extraordinary nature.
There, finally done. Sorry it took so long, hope it produces interest, instead of annoyance.