What made the English the dominant power of the British Isles?

To answer this question, you first have to answer another complex question: Who are the English?

This question turns out to be quite complex indeed because to this day scholars are unsure whether to subscribe to an invasionist/migrationist view or a diffusionist view in regards to the Britons, the Celtic people of Great Britain (excluding Scotland) which are one of the early peoples which preceded the English. To give a rough explanation, the former would mean that the Briton culture came to Great Britain from the continent with its people, whereas the latter would mean that contact with the continent transformed the existing people on the island into the Briton culture.

So, who were these people that were already in Great Britain? Modern humans first reached Britain around 42,000 years ago, but during the last glacial maximum between 25,000 and 15,000 years ago, the island is generally believed to have been unoccupied again. People may have begun occupying the island immediately after the last glacial maximum, but by 9600 BC there was certainly an established population. To give these people some context, some Mesolithic postholes were dug in the Stonehenge area around 8000 BC, so ritual activity had certainly begun.

Over the years prior to the Britons, the cultural changes are generally believed to have happened within a more or less consistent population. For example, 4000 BC brought the Neolithic culture, and the Bronze Age culture arrived about 2300 BC. By 800 BC, the Briton Iron Age had begun in Britain, and this population expanded to about three or four million by the first century BC, mostly concentrated in the south.

^{Update 4/23/2018: It looks like the Neolithic culture that built Stonehenge may have been replaced by a Bronze Age invasion, after all. See Ancient-genome study finds Bronze Age ‘Beaker culture’ invaded Britain - Ewen Callaway in Nature.}

One possible argument for the diffusionist theory is that the Briton druids are understood to have made use of Stonehenge, which was built prior to the Britons. It is certainly possible that invaders made use of the existing religious monument, but it is perhaps somewhat more likely that there existed at least a strain of continuity between the Stone Age, Bronze Age, and Iron Age peoples of the island.

Some of the Iron Age Britons may have been the first to call themselves British, or Pretanoi. The Greek Pytheas "discovered" Britain in c. 325 BC. This event marked the beginning of the expansion of the Greco-Roman world northward in Europe. Between 200 BC and AD 43, the Germanic-Celtic refugees from Gaul began migrating to Britain, as the Romans expanded into Gaul. The southern tribes started to become more Romanized, and it is this influence of Germanic and Roman culture which began the transition of the Britons into the English.

By AD 40, the Roman Empire had conquered southern Britain, and the period known as Roman Britain began. It is the Romano-British culture which gave rise to the early legends which would later be transformed into the Arthur myth. After the Romans withdrew from Britain in about AD 410, the Germanic Anglo-Saxons began their series of invasions of the island. As Wikipedia puts it:

Traditionally, it was believed that a mass invasion by various Anglo-Saxon tribes largely displaced the indigenous British population in southern and eastern Great Britain (modern-day England with the exception of Cornwall).

However, recent genetic studies show that there was not really a mass displacement at all. Rather, the relatively small groups of Anglo-Saxon invaders had a large cultural impact on the existing British population. The most notable theory to explain why the Anglo-Saxon culture had such a profound effect on the indigenous people is that the political and economic dominance of the Anglo-Saxons made their culture more desirable. Of course, what exactly gave the Anglo-Saxons an advantage is more complex, and I'll leave that up to the reader to pursue further (more information is available in the linked Wikipedia articles).

By 1066, when William the Conqueror took the English throne, the English culture was already well established, and William's advancements--such as new castles and a census--served only to strengthen the fledgling nation.

The answer to your question, then, is that the English are likely dominant because they are probably composed of the steadily growing population sprung from the more or less original inhabitants of the island. Wales and Scotland, then, are the exceptions, and they deserve further scrutiny.

Some of the Welsh tribes made peace with the Romans, and thus were able to maintain some of their identity through the Roman period. The Anglo-Saxons were not able to penetrate all the way into Wales, possibly because the Romans had not built as much infrastructure there.

The Romans were never able to invade much of Scotland, possibly because it was hard to fight the tribes in the Highlands. Eventually the Romans gave up their land in southern Scotland, and the territory was dominated by native tribes, such as the Picts in the north. After the Romans withdrew from Britain, the Gaels from Ireland invaded Scotland in the west, and the Picts withdrew to the east. Eventually, gaelicization (the influence of the Gaels), transformed Scotland, and the Gaelic and Pictish crowns merged. The strength of this new kingdom is what kept Scotland independent from the English in the south, and that is another topic that deserves further study by the reader.

Wikipedia References:

• History of the British Isles
• English people
• Anglo-Saxon settlement of Britain
• History of Wales
• History of Scotland