At the time of Roman settlement, would there have been any indigenous non-Celts left in Britain?

Being as this deals mostly with geographic areas of the world where history had not yet started, you might have better luck getting answers to this question by placing it in an appropriate subreddit, such as /r/askanthropology.

Seeing as the Romans began the history of the British isles by writing about them, and that they did not write about all the cultures existing on the isles at the time of their conquest, and they also did not always distinguish meaningfully between different "barbarian" cultures, historians would not generally be equipped with the knowledge base to accurately answer this question as much as anthropologists would.

EDIT: As a quick overview, to help you frame the question better in your mind:

All cultures before the arrival of the Romans were prehistoric, although there were some historical records of the area before the Romans due to Greek writings.
The islands have been inhabited by various species in the Homo genus for hundreds of thousands of years, and by Homo sapiens for at least 25,000 years. The Younger Dryas, a relatively brief period of intense cold, may have forced all hominids that remained there out until it ended, although there is fringe speculation, unsupported by archaeological evidence, that not only would other cultures besides the Celts have survived for a long time after the arrival of the Celts, but perhaps even other species of hominids. Before this, all of the cultures (including the non-human cultures) were paleolithic.
When the Younger Dryas ended, around 9600 BCE, it is certain that new cultural and/or ethnolinguistic groups moved in with mesolithic cultures.
By 4000 BCE the mesolithic cultures had been replaced by neolithic. Whether this was through technological adoption or by displacement has not yet been determined.
Around 2200 BCE the Bronze Age began in the British Isles, and this is the last cultural period before the arrival of the Celts. It was these people, of the Beaker pottery culture, that are the most likely to have the possibility of still existing when the Romans arrived. It is technically possible that members of any of the other previous cultural groups could have survived in isolated pockets - after all, there are members of neolithic cultures that still exist today within modern industrial states - although there is not currently any conclusive evidence that I am aware of for that.
The arrival of the insular celts roughly coincides with the beginning of the Iron Age in the British isles, around 750 BCE. They had about 700 years of dominance on the islands to wipe out other cultures before the arrival of the Romans. The only archaeological evidence I am aware of that indicate differences from the insular celtic cultures that dominated Britain at the time of Roman arrival seems to match up with continental celtic cultures.