How much did the migration of Saxons, Jutes, and Angles to England change the ethnic make up of England? Did the Saxons mostly replace the native Celtic population of England or did they merely become the ruling class?

Anglo-Saxon culture replaced the native Brittonic culture, but apart from some wars in the 400-500s, there was no extermination. It may be of some interest to you to know that the pre-Celtic population of the British Isles is more-or-less unchanged even today. The pre-Indo-European culture was extinguished by the Insular Celtic culture, which was extinguished by Anglo-Saxon culture in turn; but the descendants of the original peoples always constituted a majority and their genetic heritage is intact.

West. "Why did the Anglo-Saxons not become more British?" English Historical Review. Oxford University Press. 2000. PDF warning. Also beware her shitty etymology of Old English wealas.
Matasovic. "The substratum in Insular Celtic." Journal of Language Relationship. 2012. www.jolr.ru/files/(101)jlr2012-8(160-164).pdf PDF warning.
Wade. "English, Irish, Scots: They’re All One, Genes Suggest." New York Times. 5 March 2007. PDF warning. This is for laypersons to read.
Campbell. "Geographic Patterns of Haplogroup R1b in the British Isles" Journal of Genetic Geneology. 2007. PDF warning.