In the past, it's been said this publication was held up because reviewers had problems with the dating and attribution of the various clusters by the authors. I'd presumed such reviewers must be anti-migrationist holdouts motivated by politics. But now that I've read the paper, I can't say I'm impressed with the authors's methods or conclusions.
An example of their reasoning:
After the Saxon migrations,the language,place names,cereal crops and pottery styles all changed from that of the existing(Romano-British) population to those of the Saxon migrants. There has been ongoing historical and archaeological controversy about the extent to which the Saxons replaced the existing Romano-British populations. Earlier genetic analyses, based on limited samples and specific loci, gave conflicting results. With genome-wide data we can resolve this debate. Two separate analyses (ancestry profiles and GLOBETROTTER)show clear evidence in modern England of the Saxon migration, but each limits the proportion of Saxon ancestry, clearly excluding the possibility of long-term Saxon replacement. We estimate the proportion of Saxon ancestry in Cent./SEngland as very likely to be under 50%, and most likely in the range of 10–40%.
A more general conclusion of our analyses is that while many of the historical migration events leave signals in our data, they have had a smaller effect on the genetic composition of UK populations than has sometimes been argued. In particular, we see no clear genetic evidence of the Danish Viking occupation and control of a large part of England, either in separate UK clusters in that region, or in estimated ancestry profiles, suggesting a relatively limited input of DNA from the Danish Vikings and subsequent mixing with nearby regions, and clear evidence for only a minority Norse contribution (about 25%) to the current Orkney population.
We saw no evidence of a general ‘Celtic’ population in non-Saxon parts of the UK. Instead there were many distinct genetic clusters in these regions, some amongst the most different in our study, in the sense of being most separated in the hierarchical clustering tree in Fig.1. Further,the ancestry profile of Cornwall (perhaps expected to resemble other Celtic clusters) is quite different from that of the Welsh clusters, and much closer to that of Devon, and Cent./S England. However, the data do suggest that the Welsh clusters represent populations that are more similar to the early post-Ice-Age settlers of Britain than those from elsewhere in the UK.
The observation (Fig.2 and Supplementary Table 4) that particular European groups(for example, GER3, FRA12, FRA17) contribute substantially to the ancestry profiles of some, but not all, UK clusters strongly suggests that at least some of the structure we observe in the UK results from differential input of DNA to different parts of the UK: the absence in particular UK clusters of ancestry from specific European groups is best explained by the DNA from those European groups never reaching those UK clusters. A critical observation which follows is that groups which contribute significantly to the ancestry profiles of all UK clusters most probably represent, at least in part, migration events into the UK that are relatively old, since their DNA had time to spread throughout the UK. Conversely, groups that contribute to the ancestry profiles of only some UK clusters most probably represent more recent migration events, with the resulting DNA not yet spread throughout the UK by internal migration. ‘Old’ and ‘recent’ here are relative terms—we can infer the order of some events in this way but not their absolute times. Although we refer to migration events, we cannot distinguish between movements of reasonable numbers of people over a short time or ongoing movements of smaller numbers over longer periods.Thus, their estimate of Anglo-Saxon admixture comes from picking which of their "European groups" -- which do not, of course, represent actual ancient European groups but are merely synthetic clusters generated on data from present-day continental Europeans -- to associate with Anglo-Saxons. Going with the authors's approach, estimates of the Anglo-Saxon contribution can range from 10% to over 50%, depending on what one deems "likely".
Personally, I deem it unlikely they're picking up any signals dating back to "the early post-Ice-Age settlers of Britain". I would guess that most of their clusters are influenced by more recent patterns of migration and isolation than the authors tend to assume. More credible estimates of, e.g., Anglo-Saxon admixture in Britain will await sequencing of large amounts of ancient DNA.