When did Anglo Saxons start speaking English?

There are three phases of the English language: Old English, Middle English and Modern English. Old English is anything before 1066 (and a little after) and occurs in a range of dialects, most commonly Old West Saxon but also Old Mercian, Old Northumbrian, Old East Anglian. Middle English is post-Conquest and ends around the 16th Century, most famous for Chaucer but also widespread in other uses. Modern English starts with Shakespeare and the King James Bible and survives to today.

It is a moot point as to when this became a single language and when it stopped being comprehensible to related languages on the Continent such as Old Frisian, Old Saxon, even the Scandinavian languages. The Old Saxon poem Heliand can be translated with knowledge of Old English and it would appear that there was only a slight language barrier between Anglo-Saxons and Vikings.

So the Anglo-Saxons always spoke English, which means the language of the Angles. The Angles lived in Europe somewhere south of Denmark and most of them appear to have migrated to the British Isles, settling along the east coast and penetrating far inland, forming kingdoms such as East Anglia, Northumbria and Mercia. The Saxons settled the south of what is now England; Sussex, Essex, Middlesex and Wessex. Other peoples came with them in smaller numbers: the Jutes in Kent, Hampshire and the Isle of Wight and a good sprinkling of Franks and Frisians too.

Bede, writing about AD 730 considered there to be a "gens Anglorum" which included all of these people and he seems to consider them as speaking the same language. If you had to give a date, it would be then, but even that would be misleading.