I think you don't see what I mean. My concern is not with the content but with the terminology. The word 'basin' has three definitions: a structural/tectonic one, a sedimentary one and a geomorphological one. All three seem in some way to apply to the London Basin (and the exact boundaries of the London Basin may differ depending on the definition!). The word syncline only has one definition, a structural one. The original sentense called an apple a kind of pear. 
I agree with you that the synclinal structure is essential and should be discussed, preferably in the header of the article. But a syncline is in itself not a sedimentary basin. It is a structure on its own. The occurence of a syncline at the core of a basin is, however, usually not accidental. When strata are folded, someone may of course think the synclines will form a basin. In nature though, uplift is never quick enough to create structural basins in this way, certainly not at the scale of the London Basin. Erosion will remove the anticlines and sediments will fill the synclines almost instantly (instantly in geologic time). As folding continues, so will erosion and sedimentation. Eventually a structure will form which is both a structural syncline and a sedimentary basin (a basin in the sense of: a place where sediments have accumulated). This model is called syn-tectonic deposition (or syn-depositional folding). As far as I see, this may well be the case for the London Basin, but I am not sure since we haven't found conclusive information in the sources yet.
There is another way how the strata can be folded into a syncline: pre-tectonic deposition (or post-depositional folding). It is important to find out which of the two models applies. The test will be that in a syntectonic basin, the strata will wedge out in the direction of the edges, while in a pre-tectonic basin, the strata will on average have the same thickness everywhere. The outcrop pattern on the geologic map doesn't show the difference, because the map is not detailed enough.
That the older strata of the Chalk are also folded by the syncline is clear, but I don't see what that has to do with the basin on top of them?