I'm also not an expert in this field, but I agree that the conclusions presented here appear at first glance to be much too strong. For one, we should all be highly skeptical of meta analyses to begin with - there is every incentive to report "significant" results, and a documented tendency for such studies to be preferentially published, so the current literature should be enriched with respect to rejections of the null.
Also, did I miss it, or is there zero discussion about how the discussed publications were selected? A quick google search turns up lots of additional null results.
The authors also do make some correct criticisms of common statistical mistakes. For example, it is true that failure to reject a null hypothesis is not evidence for that null hypothesis in a frequentist framework. Even so, we should consider the power to reject the null, which depends on:
1. The practical or clinical effect size
2. The sample size
Even if we do detect statistically significant effects of EMF/WiFi, that in no way implies that the results are of practical significance. The cited studies do have small samples, but no discussion is given to the magnitude of effect which would be expected to have been detected.
They also note that previous studies should have published their data and p-values, which is true.
My takeaway is that more information is needed, and that we should hear from someone with a good grasp on all the relevant literature to see how representative these studies are. Also, this kind of thing would be clearer if we were all Bayesians, and published our data :P