It appears that it would be true only by coincidence. Inherited surnames in England are dated as starting from the 12th C (some 150 years after the conquest of the Anglo-Saxons), according to P. H. Reaney and R. M. Wilson, A Dictionary of English Surnames, 3rd ed., pp. xlvi et seq. That book is a good source. P. 156 has header name English, Inglis, Inglish. They derive the word from Old English to refer to Angles as opposed to Saxons, but not the surname: "a meaning not to be considered for the surname". They suggest

• Scots Inglis (one of the early citations they give is from Black, Surnames of Scotland), "denoting an Englishman as opposed to the Scottish borderer or the Celtic Scot"
• in northern England, an Englishman living among Strathclyde Welsh
• an Englishman in predominantly Welsh communities on the Welsh border
• in the Danelaw, "to a newcomer who did not share the common Danish origins with its flourishing Danish customs and names"
• "For some generations after the Conquest an official distinction was made between Angli and Franci, the native defeated English and the conquering Normans, and this may account for the name in Essex, Kent and Sussex, where it was probably at first derogatory".
• Someone l'Engois was in Paris circa 1300, which surname might have migrated back to England.

A weaker argument: One of the purposes of a surname was to distinguish one person from another. Originally, surnames in England were literally true -- Smith was a smith, Red was red-haired, et cetera. It would have been less useful to have a surname that could apply to a number of local people, but I think that's a weak argument: I expect that counter-examples could be found.

I also suggest that the question is ill-posed -- what would "forebears were Angles" mean, given that the number of ancestors could double in each generation back? In any event, I don't know when they stopped distinguishing between Angles versus Saxons, though R&W (supra.) do mention that as an Old English definition.