To be honest, much of this feels like very irresponsible journalism, partly on the part of the BBC and very much so on the part of Science alert.
If you're looking for an accessible resource to what the paper does, the cover piece on APS Physics and the phys.org piece are much more sedate and, I think, much more commensurate with what's actually reported in the paper.
The paper itself is very moderate in its claims and it restricts itself very well, from what I can tell, to what they found: that certain radium and barium nuclei appear to be pear-shaped. Finding pear-shaped nuclei is not new (a similar paper (eprint) made the news in 2013, and was discussed on this site here), though Bucher et al. seem to have found hints of a discrepancy with theory with regard to just how pear-like these nuclei look like. This is, however, not at the level of statistical significance that would require any rethinking of the theory at this time.
It is important to note that pear-shaped nuclei are indeed consistent with the Standard Model of particle physics. Pear-shaped nuclei are a bit of a problem because their shape has a direction, that is, you can draw a vector that starts at the flat end and points toward the pointy end (call this the pear vector $\vec P$). (The alternative, a rugby-ball-shaped nucleus, has an axis, but no preferred direction on this axis.) Because of symmetry considerations, this pear vector $\vec P$ needs to be on the same axis as the nucleus' spin $\vec S$, but these symmetries don't tell you which way they have to point, so you get two different versions of the same nucleus:
Mathematica code for this image through Import["http://goo.gl/NaH6rM"]["http://i.stack.imgur.com/HLcYp.png"], CC BY-SA with attribution to this page.
In a theory of nuclear physics that is mirror-symmetric, then both of these nuclei need to be completely equivalent (and, particularly, have the same energy), because they are mirror images of each other. What this paper finds (and what Gaffney et al. found in 2013) is that there are nuclei where this is not true, and these two nuclear states have different energies: the ground state is the "pear" state, and not the "anti-pear" one.
Fortunately, this is not a problem: in fact, we've known since 1956 that nuclear physics is not mirror symmetric, i.e. it is not invariant under the parity operator $P$. Fortunately, though, there is a related symmetry that takes up the slack, and it is charge conjugation symmetry $C$, which takes matter to antimatter and vice versa. Much of the Standard Model, including a lot of nuclear physics experiments, is $CP$ symmetric: if you take a mirror version of the experiment, and on top of that you swap out all particles for their antiparticles, then the physics is the same.
However, $CP$ violations are still compatible with the Standard Model and have already been observed experimentally. On the other hand, the known $CP$ violations are not really enough to explain the matter-antimatter balance in the universe (a.k.a. the baryogenesis puzzle), which is not explained by the Standard Model, so any beyond-the-SM $CP$ violation is a good place to look for solutions to the baryogenesis problem.
There is also a bigger, stronger symmetry, which happens when you combine $CP$ inversion with time inversion $T$ to get what's called the $CPT$ transformation. Because of very basic facts about spacetime, all reasonable physical theories must be $CPT$ symmetric. This is one of the reasons $CP$ violations are so interesting: they point to microscopic violations of time-inversion symmetry.
So how does the paper at hand relate to all these generalities? The authors have confirmed the existence of $P$ violations, already observed, and they have found hints that these violations - the peariness of the pear-shaped nucleus are stronger than the existing theory. However, they are not comparing against ab-initio theory (i.e. they compare against approximate theoretical models, so the fault could be in the approximations they made) and, to quote from the paper's discussion,
the large uncertainty on the present result does not allow one to elaborate further.
So how did we get from there to time travel? That's where you need a large amount of journalistic 'creativity' for the joins to work. The APS Physics piece is clear and to the point, and it makes a good show of understanding the limitations of the paper.
On the other hand, the press release, from the University of West Scotland, is already pretty breathless. They quote at length from (presumably a direct interview with) Dr. Scheck, but I think his claim that
Further, the protons enrich in the bump of the pear and create a specific charge distribution in the nucleus, which shouldn’t be there according to our currently accepted model of physics.
is stretching a fair bit the pretty modest claims of the paper (cf. supra). Dr. Scheck goes on to claim that
We’ve found these nuclei literally ‘point’ towards a direction in space. This relates to a direction in time, proving there’s a well-defined direction in time and we will always travel from past to present.
but, same as John Rennie, I'm struggling to see how he connects the (known, possibly stronger than accepted) $P$ violation they found to a confirmation of a $T$ violation, let alone relating that to time travel.
Onwards from this, the BBC piece offers very little above the UWS press release, and this is a pretty bad sign - while the BBC does often have great science content, this piece is essentially a redigested press release with the hype turned up one notch, and I can't resist pointing out that things like this have been pointed out as one cause of the bigger problems science has at the moment.
The Science Alert piece (also syndicated at RedOrbit and Business Insider), on the other hand, goes a bit further along on the dodgy journalism side. In particular, it misidentifies the Scheck interview as with the BBC instead of a press release, and it quotes from a couple of stories from last year, but makes it sound like they are reaction quotes from experts interviewed about the recent paper. Other than that, it feels like a re-re-digest piece with the hype turned up three notches.
If you want the hype, then, go to the press release - that's its job. If you want a sober assessment of the implications of the work, go to the APS Physics piece or the phys.org one, which make it clear that there are few implications yet beyond nuclear physics - if indeed the result stands the test of an accurate measurement.