There is a lot of controversy over the british iron age peoples not being true celts. Although they do have some traits with mainland tribes, was the british celt a made up image and if so who were the real Iron age british culture/people?
The "native" British people, who predated the Celts, Romans and Saxons, arrived around 15,000 years ago, as the last Ice Age ended. In The Tribes of Britain by David Miles, it is argued that around 80% of genetic material of modern (white) Brits can be traced back to this people. In 1995, a 9000 year old body was found in Cheddar Gorge, and genetic tests found a common maternal ancestor with many of the local residents.
At that time Doggerland had not yet flooded (which started around 6,500 BC) so migrants would have been able to simply walk across from what is now "the continent". Archaeological evidence shows settlements in Nottinghamshire, Norfolk and Devon.
The culture of this prehistoric people is pretty much completely unknown. We don't know anything about their language or religion. There are, however, some words in modern English which may trace back to the language of these peoples. In Ackroyd's History of England Volume I, it's mentioned that "London" and "Kent" have no known etymology from any Celtic, Germanic or Romance language, and so theorised that they come from the language(s) of prehistoric Britain.
Archaeological evidence, such as chariot burials, for Celtic culture in Britain date to around the 6th century BC. Celtic culture and language appears to have dominated during this period, but as mentioned above, the Celts appear to have had little genetic impact on the British population. The extent to which the Celts displaced/merged with the pre-historic people is therefore not clear. It's certainly not the case that Britain could be said to be completely Celtic: the Roman writer Strabo distinguished between Celts and Britons (Collis' The Celts: Origins, Myths and Inventions).
It's also worth mentioning that these pre-historic Britons were not as backwards or isolated as many suppose. They were a part of the ancient trade network which stretched from Britain to the Middle East. In A Splendid Exchange: How Trade Shaped the World, Bernstein notes how tin from Cornwall and Brittany was finding its way to Sumerian Ur as early as 2800BC, where it was used to make bronze.
An examples could be a tribe in Pre-Saxon Scotland called the Picts. They were thought to be of non-Indo european origin. They're mostly assumed to speak a Celtic langauge and have a lot of Celtic influences from neighboring tribes. They combined kingdoms with another Gaelic kingdom to form Scotland around 840 (843 to be exact. It's a traditional year like how England has 927).