There is no short answer to this, but I'll do my best to summarise some of the key points.

Step 1. Britannia, Hibernia and Caledonia

These were the names given by the Romans to the territories that are now England & Wales, Ireland and Scotland respectively.

Prior to the Roman invasion of Britannia in 43 AD, the four 'countries' were made up of lots of separate Celtic, Proto-Celtic and Belgic tribes. Unfortunately we have very little information about any of these tribes, as they were all seemingly illiterate, and only described in passing by Roman writers. They did, however, appear to have a common culture, evidenced by their collective interest in standing stones, rimmed vessels and burial mounds.

When the Romans invaded, they conquered and subordinated the tribes of Britannia (England & Wales), but failed to advance into Pictish territory in Scotland, and decided to end their campaign by building a defensive ha-ha rampart (Hadrian's Wall) between Britannia (England) and Caledonia (Scotland). This border would later move slightly north, forming roughly what is still today the border between England and Scotland.

So, Scotland and Ireland were, to be blunt, largely ignored by the Romans, while England and Wales started to develop Roman roots. This is the first period in which the cultural divisions between the four counties started to emerge.

Step 2. Nordic vs. Saxon tribes

After the Romans deserted Britannia in the 4th Century to fight wars along the Empire's eastern border, their territory came under attack by multiple different tribes, but two sets in particular: the Vikings and the Anglo-Saxons.

The Vikings swept through large parts of Northern Scotland, Northern and Eastern England and Eastern Ireland, as well as the Isle of Man. Many of these parts of the islands came under a new form of control called 'Danelaw'. The Saxons, meanwhile, settled mainly in the South of England (except Cornwall). Their period was frought with battles, both against the Vikings invading from the north, and against other Saxon tribes wrestling for control.

Wales, meanwhile, was left alone in this period, devoid of all Saxon and Nordic influences. In the absence of Roman hegemony, Wales was free to continue its Celtic culture, and it is at this period that the 'modern' Welsh language was born.

Because of constant power struggles, it's difficult to determine who, what and where the king/queen of each country was during this time. However, it is fair to say that each region developed independently depending on who it was governed by. For England especially, it was the Anglo-Saxons who held power until the reign of Harold Godwinson.

Step 3. The Normans

The Normans, hailing from Normandy in Northern France and under the command of William 'The Conqueror', invaded England in 1066 over a dispute about the lineage of the English crown. William slayed Harold on the battlefield which ushered in the age of Norman control. The Normans were a brilliantly advanced society, and one of the first things they did was begin to take stock of their land (recorded in the Domesday book), allocate land to various earls, lords and bishops in order to balance their power, and institute systems of law.

At first, the Normans concentrated their power on the former Anglo-Saxon kingdoms of England. Later, they started to penetrate into Southern Scotland and Ireland. In the latter, they ruled indirectly via a system of Lordship until Henry VIII became the first King of Ireland. Multiple attempts to conquer Wales were unsuccessful, leaving the country divided between some English-ruled parts, governed by a Principality, and others, which were independent. It was only in 1535 that England and Wales agreed to unite, becoming the Kingdom of England & Wales.

Throughout this period, the lineage of each nation's royal family starts to grow complicated and messy, but what is important is to note the sweeping Norman influence that starts to appear in each one. This becomes especially significant in the early 17th century when…

Step 4. Queen Elizabeth I dies childless

… which briefly causes a succession crisis. This is resolved by tracing Elizabeth's ancestry back, which reveals her cousin and the first in line for the throne of England is, ironically, King James VI of Scotland. Yes, the man already on the throne of Scotland is the rightful king of England and is duly crowned, giving him the confusing title King James I and VI of England and Scotland.

One of James' most significant policies was the Ulster Plantation, in which he secured a vast swathe of land – almost 1/4 of the island – in Catholic Ireland, to be given to Presbyterian/Protestant settlers from Scotland and Northern England as farmland. This was the precursor to the creation of Northern Ireland.

If you're confused by the criss-crossing lineages here, so was everybody else in England, Wales and Ireland who, at this point, were asking 'who actually governs us?' And what's the best way to answer a question like this? Why, it can only mean one thing…

Step 5. The Civil War

This was a whole thing that I won't get into. What's pertinent to mention is that the war was fought over which system of governance the countries would use henceforth (Monarchy vs Parliament) and also divided into three separate theatres: the English, the Scottish and the Irish Civil War. Although, naturally, the English was the most significant, as this was where the leader of the Royalists/Cavaliers (King Charles I) and Parliamentarians/Roundheads (Oliver Cromwell) spent most of their time duking it out.

The "winner" (if indeed there ever is a winner in a civil war) was Cromwell, who duly beheaded Charles I, brutally subjugated Ireland and Scotland, and ushered in a puritanical dark age which was so universally unpopular throughout the kingdoms, that after only a couple of years the people were begging for Charles I's son, Charles II (formerly King of Scotland, just to add further confusion) to return from exile and re-start the monarchy.

Upon his return, Charles II is crowned King of England, Scotland and Ireland.

Step 6. The Darien Scheme, or how Scotland shat the bed

There were various precursors to the Act of Union in 1707, many of which stemmed from the Reformation and the Civil Wars. The cultures of the three main territories – England (& Wales), Scotland and Ireland – were starting to grow closer. English had solidified itself as the de facto language of both islands.

However, the official uniting of the kingdom was a direct consequence of the failed Darien Scheme, something which – as a Scot – I can only look back and laugh at.

At the beginning of the Age of Empires, and while seeking their own share of global prestige (and wanting especially to emulate the success of their English neighbours), Scotland invested in a scheme to colonise a large tract of land in Central America. The scheme was an unmitigated failure for various reasons (disease being a key one) and essentially drove the whole country to bankruptcy, which forced Scotland to pursue unification with England. This, in turn, became the Act of Union in 1707.

There is a final chapter to this story which is the Irish Civil War and Irish Independence movement, but that is beyond the scope of my argument, which is to answer the question of how each of these four countries developed a unique and distinct cultural identity which is still represented in the fact that they are both officially and unofficially four (or rather five, including Northern Ireland) different countries today. The 'creation' of Northern Ireland is based in the Ulster Plantation which pre-dates the Act of Union.

In summary, each country's legacy and influences can be summarised thus:

England: Proto-Celtic, Belgic, Roman, Anglo-Saxon (South), Viking (North), Norman, Tudor, Stuart influences.

Wales: Celtic, Roman, some Norman, Tudor, Stuart influences.

Scotland: Celtic, Viking (Highlands), Norman (Lowlands), Stuart influences.

Ireland: Celtic, Viking (Northern), Norman (Dublin), Tudor, Ulster Plantation (Northern Ireland), Stuart influences.