0 AD is kinda a tricky time period. You are between Caesar's invasions of Britain and the Claudian conquest so records are rather fragmentary. Its more proto-history than history. And even that really only covers southern England. Scotland and the north is archaeology. Worse still due to the loss of traditional trading routes with Gaul it seems to have been a rather unstable period.

So even today figuring out what was going on is a major challenge.

1000AD? Well probably the most available document would have been Julius Caesar's Commentaries on the Gallic War. Which wouldn't be far off your standard pop culture version of Celtic Britain.

Historia Brittonum which was written a bit after 1000AD but probably at some point during that century late 9th century simply jumps straight from Caesar's invasion to the Claudian conquest. It does however divide Britons from Scots and Picts and provides them with a suitable mythological background.

Bede's Ecclesiastical History of the English people makes the same jump although it does cover events in slightly more detail (athough it gets the number of invasions wrong).

To what extent did they identify with them?

In the case of what is now England not much. Æthelred the Unready promoted himself as king of England not King of the Britons and tended to make references to Rome or earlier English kings.

In the case of Scotland 1000AD is getting pretty close to the end point of the Picts but Gael identity is starting to shift to Scotts.

In the case of what is now wales some of more sucessful rulers seem to have been referred to as King of the Britons although to what extent that was anyone's primary identity is a seperate issue.

This is a great question. I just feel it's important to point out that there was no such thing as 0 A.D. The year A.D. 1 followed directly on from the year 1 B.C.