Yes, the strange people Dickens mentioned would have been regarded as Celts.
The French and Belgians are the Gauls, with the northern Gauls referred to as Belgae by the Romans. The first Roman to make this distinction was Julius Caesar which he does at the start of his "books" on the Gallic Wars where he states that [unconquered] Gaul was divided into the Belgae in the north, the Aquitani in the southwest and the rest was inhabited by Celtae.
Dickens must have been aware of this which is why he limited this group to the French and Belgians and didn't include the Dutch, though the ancient Belgae included the southern part of what became The Netherlands (home of the Dutch).
It had been known since the early 18th century that the languages of the Welsh, Bretons and Cornish were related to those of the ancient Celts. The first to publish this conclusion was Yves-Paul Pezron in his 1703 book 'Antiquité de la nation, et de langue des celtes'. Edward Lhuyd's 1707 'Archæologia Britannica' fleshed out Pezron's work in a more scientific fashion and added the concept that the Irish language came earlier, then a later invasion brought the Brythonic language which Welsh etc derived from.
The Irish have a legend that the first Irish arrived under the leadership of a man called Mil Espaine and that they originated in Spain. Regardless of whether this has any merit it was a widespread belief and still holds many supporters today.
Dickens Iberian strangers are therefore referencing this Irish legend which then ties in with what scholars like Edward Lhuyd were suggesting that the Irish language arrived in ancient times followed by a later invasion that brought the ancestor of Welsh and Cornish.
This later invasion can be supported by Julius Caesar who in chapter 12 of his fifth book claims the Belgae had invaded Britain and imposed themselves over much of the coastal regions.
So using medieval and ancient sources and some "new" scholarship which by the time Dicken's children's history book was written was nearly 150 years old so well established, we have the basics of his facts - the Irish coming from Iberia (Spain) and the Belgic invasion of the "French" and Belgians.
So what part do the Phoenicians play in this?
The ancient Phoenicians knew about Britain, it was a source of tin which was hugely important during the Bronze Age for being one of the main ingredients of bronze. The trader Himilco who lived around the late 6th, early 5th century bc wrote of the British Isles being made up of the main islands of Albion and Ierne (Britain and Ireland). We know this because the Romans were still writing about him centuries later as part of Roman geography books and poetry like the Ora Maritima written in the 4th century ad, 800 years after Himilco's time.
The Phoenicians were well known explorers and merchants, the most famous Phoenician city being Carthage which created a major Mediterranean Empire.
Where Dickens falls flat is assuming that the French and Belgians had to be told about the richness of the lands of Britain by these Phoencians despite living in plain sight of the British coast. Archaeology has showed that trade between Britain and continental Europe had been constant since the early Bronze Age but Dickens is obviously taping into an assumption of the time that neither the Britons nor the people opposite in Europe had anything to do with each other. So the great Phoenicians told the French and Belgians about the rich mines and other resources of Britain which made them cross the Channel and settle there along the coast - at which point Dickens paraphrases Caesar about the Belgae in the maritime districts and the natives in the interior.