It's never easy to pinpoint exactly when a specific language began, but in the case of English we can at least say that there is little sense in speaking of the English language as a separate entity before the Anglo-Saxons came to Britain. Little is known of this period with any certainty, but we do know that Germanic invaders came and settled in Britain from the north-western coastline of continental Europe in the fifth and sixth centuries. The invaders all spoke a language that was Germanic (related to what emerged as Dutch, Frisian, German and the Scandinavian languages, and to Gothic), but we'll probably never know how different their speech was from that of their continental neighbours. However it is fairly certain that many of the settlers would have spoken in exactly the same way as some of their north European neighbours, and that not all of the settlers would have spoken in the same way.
The reason that we know so little about the linguistic situation in this period is because we do not have much in the way of written records from any of the Germanic languages of north-western Europe until several centuries later. When Old English writings begin to appear in the seventh, eighth, and ninth centuries there is a good deal of regional variation, but not substantially more than that found in later periods. This was the language that Alfred the Great referred to as ‘English’ in the ninth century.
The Celts were already resident in Britain when the Anglo-Saxons arrived, but there are few obvious traces of their language in English today. Some scholars have suggested that the Celtic tongue might have had an underlying influence on the grammatical development of English, particularly in some parts of the country, but this is highly speculative. The number of loanwords known for certain to have entered Old English from this source is very small. Those that survive in modern English include brock (badger), and coomb a type of valley, alongside many place names.