When you ask about the process of combining Scotland and England into a single kingdom, it is unclear as to whether you are asking about social processes, political processes, economic processes, or some combination of all three. That said, I am going to make an assumption based on your post history that you are *more* interested in social processes than the other two, and that your question, rephrased, is something like "How were the Scots and the English able to peacefully assimilate after the Union of the Crowns since they're pretty much always characterized as historic enemies who hate each others' guts". In order to unpack this, we need to consider a couple of key ideas:
On this first point, although the Scottish Wars of Independence (1296-1328 and 1332-1357) are often painted as a nationalistic struggle against an English oppressor, these conflicts should, in fact, be viewed with more political nuance and that nuance should be extended to the people involved on either side. Anglo-Scottish conflict became recognizable as such (rather than as small individual raids across a dynamic border conducted by families on both sides of that border) following the Norman Conquest of England. In the 1070s, William I successfully subdued the rebellions against his rule in the north of England and continued further north into Scotland, where he was met by Malcolm III at Abernethy. It is important to note that William blamed Malcolm for supporting the northern rebels because Malcolm's wife, (St.) Margaret, was an exiled princess of Wessex and her brother, Edgar Ætheling still held land there; William believed that Malcolm's support of the rebellions was intended to place Edgar on the throne of England. No battle ensued at Abernethy, however, as Malcolm agreed to accept William as his overlord, though this was seen as nominal, if anything, for Malcolm raided Northumbria in 1080 in an attempt to push the Anglo-Scottish border further into England. Once more he was forced to acknowledge the overlordship of William I and he returned north to Scotland. In 1091, peace between the two crowns began to break down once more as William Rufus (son of William I) threatened the estates that William I had granted to Malcolm III for his maintenance when visiting England (it is unclear if William Rufus actually confiscated these lands or merely threatened to do so). In response, Malcolm once more marched south but refused to have the debate decided by the English barons as suggested by William Rufus. On his way back to Scotland, he was ambushed by the earl of Northumbria whose lands he had devastated in his earlier raids, and in retaliation, killed near Alnwick on 13 November 1093.
Malcolm's youngest son, who became David I, also went to war with England. This was ostensibly due to the usurpation of the throne by Stephen as David I was married to Henry I's sister and thus it was David's niece-by-marriage whose claim Stephen had upset; some historians, however, take a more cynical view of David's support of the Empress Matilda, suggesting that this support was given with the intention of demanding land in the north of England as compensation should Matilda successfully regain her crown.
In 1173, David's grandson, William I of Scotland (not to be confused with William of Normandy), called "William the Lion", supported the rebellion of the English princes against Henry II. At the Battle of Alnwick against the king's troops, William was unhorsed and captured, and sent to Normandy as a prisoner. Henry II then invaded Scotland and left an army of occupation there until William the Lion agreed, in 1174, to sign the Treaty of Falaise, which acknowledged, once again, the English king's overlordship of the king of Scots. He was also forced to pay Henry for the cost of keeping his army in Scotland and this was done through raising taxes in Scotland. If Scottish sentiment towards the English was less than favourable, then, it likely had a lot to do with being forced not only to provide for an occupying army but to then pay for that army to leave through the medium of increased taxes.
When Henry II died in 1189, William petitioned Richard I to nullify the Treaty of Falaise and this was accomplished with the Quitclaim of Canterbury, which negated all terms of the earlier treaty, including the need to continue paying back the sum that had been imposed for the English army's maintenance, which William was still paying at that point.
The relationship between the kings of Scots and kings of England after this point were mostly peaceful until the death of Alexander III in 1286 and the death of his heir, the six-year-old Maid of Norway whose ship sank in 1290 on its way to Scotland as she traveled for her coronation. In response, a number of Scottish nobles who could claim some (primarily matrilineal) descent from the Scottish royal family put forward their claims to the throne and the Scottish regency council invited Edward I of England to adjudicate these claims. Edward, seeing an opportunity to effectively annex Scotland at this time, required the political community of Scotland to acknowledge his overlordship over the Scottish crown for, he argued, how could he have any authority to adjudicate if he was not given authority over the king of Scots? He was duly acknowledged and he chose as the next king, John Baliol whose claim derived from the fact that Devorguilla of Galloway, John Baliol's mother, was the daughter of Margaret of Huntingdon, who had been the niece of both Malcolm IV and William the Lion. John Balliol, then, was the great-great-great-grandson of David I of Scotland. Another claimant to the throne, Robert Bruce, son of Robert, Lord of Annandale, was also a descendant of David I, as his great-grandmother, Isobel of Huntingdon, was Margaret of Huntingdon's younger sister. Bruce and John Balliol were, therefore, second cousins once removed.
Considering John a usurper, Bruce allied himself with Edward over John and his Comyn allies when the new king of Scots protested continued English interference in Scottish affairs. When war was declared in 1296, the Bruces were primary targets as the Comyn host first attacked the city of Carlisle, which, though held by England at the time, was a known stronghold of the Bruce family. In response, Edward invaded Scotland. The following year, Bruce broke his alliance with England and joined the Scottish rebels. Thereafter, Bruce fought against the English until the end of the first war in 1328, being crowned king of Scots in 1306, John Balliol having abdicated in 1296. War again erupted in 1332. The Second War of Independence was much more complicated than the first and included more participants than the first as well. The takeaway is that the war was caused by Edward III's feelings of humiliation, a number of Scottish nobles who were upset that their lands and titles had been bestowed upon Robert Bruce's allies following their opposition to him during the First War of Independence, and was prolonged by the intervention of France in the 1330s and 1340s as England and France embarked upon the Hundred Years' War.