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.