I have four things to say regarding this:
The paper is far too new. I'm not saying the first paper is wrong (it is in fact rather interesting), but having been published yesterday, it is far too new to be used as the basis for anything. Publication in a peer-reviewed journal is where science starts rather than ends. Wait a bit for the scientific community to confirm or reject the claims in this paper.
The two effects might be completely unrelated.
If they are related, you might be confusing cause and effect. It could be that whatever causes the climate oscillations in turn causes the oscillations in the inner core's rotation rate.
What the authors of the paper in question are reporting to have observed is a small differential rotation between the mantle+crust and the inner core. It could well be that the inner core rotates much more smoothly than does the mantle+crust. In other words, they're seeing variations in the rotation rate of the mantle+crust.
We know as a fact that the Earth's rotation rate is gradually slowing down. Scientists see this far in the past in tidal rhythmites locked in stone. More recent signs were ancient astronomers/astrologers (the two were the same "science" up to Kepler's time) who recorded solar eclipses at specific cities. Those solar eclipses would have happened elsewhere on the Earth if the Earth maintained a more or less constant rotation rate.
Thanks to atomic clocks and long baseline interferometry, scientists readily see seasonal changes in the Earth's rotation rate. Snow falls on the northern parts of the North Hemisphere in winter and melts in the summer. That represents transfers of angular momentum between the oceans, the cryosphere, and the crust. These variations are easy to see.
Precise measurements of the Earth's rotation rate also show what appears to be longer term oscillations in the Earth's rotation. It appears that there is a multidecadal oscillation in the Earth's rotation rate that is correlated with the 60 to 70 year Atlantic Multidecadal Oscillation (AMO). Since very good data on the Earth's rotation only goes back 60 to 70 years, it's a not quite conclusive. The same goes for the AMO; very good meteorological data only goes back 100 years or so. The cause of the AMO is debated. The teleconnections between it and other climate observations are also debated. However, the AMO does appear to impact climate, and climate in turn appears to impact Earth rotation. (We know climate as a fact that impacts Earth rotation on a yearly basis.)
The Earth comprises multiple rotating parts: The atmosphere, the cryosphere, the oceans, the crust and mantle, the liquid outer core, and the solid inner core. These different parts exchange angular momentum with one another (and also with the Moon). Except for the seasonal variations and the long term secular transfer to the Moon, teasing out cause and effect in the variations of the Earth's rotation rate is a non-trivial exercise.