Leaving aside any questions over the validity of the pedigree, this is an example of where naming conventions of the time do not quite map between countries or to current naming conventions. By this era, England was using heritable surnames (typically from ~1400 onwards) but in Wales the convention remained use the given name of the father as the second name of the son, thus 'Morgan ap William' literally translates as 'Morgan son of William', 'ap' being a contraction of the word 'mab'. This convention remained in some places into the C18 and tended to be replaced by simply converting the last such generation into a phonetic equivalent of 'son of XXX's', thus 'son of John's' became 'Jones', 'son of William's' became 'Williams' etc. It is reasonable to assume that, on moving to London, Morgan would have followed this convention.
However, which name to use as LNAB is slightly less clear. It is rather like the Norman convention of naming after places, incorporating 'de' which is omitted within WikiTree. Morgan would never have been known as 'Morgan William' but equally the inclusion of 'ap' may not be acceptable.
As an aside, the equivalent patronymic naming convention is still used in Iceland but it simply incorporates the context within the suffix of the name, e.g. William's son would be Morgan Williamson and if Morgan happened to have a daughter she would be Morgansdottir.