That’s an interesting question and there are a variety of dates that one could point to, depending on the particular region of Scotland.

In the case of south-eastern Scotland around Edinburgh, the Lothians and borders, Old English replaced the local Brythonic language when it was incorporated into the Anglo-Saxon kingdom of Bernicia (later Northumbria) in the mid 7th century. By the mid 8th century, Northumbria was well established and was even extracting tribute from the Kingdom of Strathclyde in south-western Scotland. This area fell under Viking influence and then under the control of the Kings of Wessex before Scottish control was established in the 970s during the reign of King Kenneth II. At this point, the MacAplin, Gaelic speaking dynasty ruled and so the court language was Gaelic, but Old English was still spoken in the south-east and Brythonic in the south west.

The next key turning point is the reign of King David (1124-1153). His father, Malcolm III, had married Margaret of Wessex and David spent much of his youth at the court of Henry I, marrying Henry’s sister, Maud. Even as heir apparent, he began inviting Anglo-Norman nobles to settle in southern Scotland. Indeed, the first surviving charter from his reign details a large land grant to the de Brus family (a Norman family later to become famous as Bruce). He also established the first “royal boroughs” in Scotland including Berwick, Perth and Aberdeen. The lingua franca in these trading towns was Middle English and it became the language of commerce, although Scottish kings were fluent in English, Norman French and Gaelic throughout the 12th and 13th centuries, with some closer to England and others closer culturally to their Gaelic roots.

By the 1370s, Middle English had grown to the extent that Scottish chronicler, John of Fordun, wrote “The manners and customs of the Scots vary with the diversity of their speech. For two languages are spoken amongst them, the Scottish [Gaelic] and the Teutonic [English]; the latter of which is the language of those who occupy the seaboard and the plains, while the race of Scottish speech inhabits the Highlands and outlying islands... The Highlanders and people of the islands... are a savage, untamed race, rude and independent.” By this point, Gaelic was on the retreat and, throughout the Stewart period, was eclipsed. In fact, it began to be called “Irish” by lowlanders, who referred to their own language (now known as Scots) as “Scottish”.

Scots had become the main language in the towns and at court, whereas Gaelic was associated by many with barbarism. Events such as the response to the Jacobite rising of 1745 and the Clearances further weakened Gaelic-speaking society and depopulated the Highlands.

As for how English replaced Scots, many enlightenment figures of the 18th century, such as Adam Smith and David Hume, popularised the notion that Scots was a bastardised form of English and promoted “proper English speech” by which standard English began to replace Scots. By the Victorian period, Scots was already falling out of favour and was increasingly spoken in rural areas, whereas Standard English became the prestige dialect.

Tldr: Scots had replaced Gaelic in the lowlands by the mid 14th century and increased in influence over time as it was the prestige language.

Main source: Scotland: A New History, Michael Lynch, 1992