Celts and Germanic tribes already lived in Britain by the time of the Norman conquest, so it would be difficult (but not impossible) to completely destroy the languages that were already there. I say "not impossible" because the Romans were extremely successful in destroying the Celtic languages of the Gauls and replacing these previous languages with vulgar Latin. But I think the POD of the Norman Conquest, almost a thousand years later than the Roman destruction of Celtic Gaul, is a bit late to change the syntactical movement of Old English.
Vocabulary is a different story. Lots of words have survived from Anglo-Saxon to modern English, many of which I can't say in polite company. The replacement of many of these words with Norman analogues relegated the Anglo-Saxon words to a derogatory status. But at one time these words were perfectly acceptable words to describe bodily functions and sexuality (among the more famous Anglo-Saxon words that have survived to today). Germanic syntax has remained strong within the English language, even to this day, as even modern English is influenced more by German paradigms than Romance paradigms. But perhaps the Normans might have been more successful in "stamping out" Anglo-Saxon analogues and replacing them with Norman words. The beauty of the English language is the flexibility of its vocabulary; to rob the language of its Germanic-derived words would rob us of our ability to swear!
