It is not correct to say that English common law and equity was merged in the judicial reforms of the 1870s; they remained distinct areas of law. What changed is that they were administered by courts with both common law and equitable jurisdiction, but they were to be fire-walled in doing so. As late as the 1990s there was fierce debate in legal academia over whether law and equity had fused or not. Which is where the debate belongs: in practice, whether they have fused or not doesn’t matter - the courts do what they do regardless.
So, the short answer to the question is, they haven’t - law and equity are still their own beasts and courts treat them as such.
However, if your question is about how colonial outposts of the British Empire dealt with giving law and equitable jurisdiction to the same court, well, it varied.
The British Empire never had a unified legal system but instead consisted of seperate jurisdictions each with their own unique way of doing things. In colonies that were terra nullius (e.g. America, Canada, Australia), that legal system was English at it’s root but in other colonies (e.g. India, South Africa) the existing legal system remained which then got overlaid with English law. At the same time, colonies were not created in a vacuum, they had colonial charters that set out how they were supposed to work, with greater or lesser autonomy.
This 1971 essay sets out how the various American colonies dealt with Chancellery (also known as chancery or equity) courts, if they did at all. One of the main distinctions between the Law courts and the Chancellery courts is their constitutional basis - the Law courts were creatures of the common law, they were “of the people”; while the Chancellery court was a Royal Prerogative, they were the King descending from on high to do justice and mercy. This is important because the early days of the American colonies coincided with some major constitutional events in England - the Civil Wars, the Interregnum, the Restoration, and the Glorious Revolution which ultimately established the supremacy of Parliament.
However, while that was going on, the American colonies were still in their first few generations and frontier justice was still pretty rudimentary. Equity deals with pretty complex subjects that arise in relatively complex societies and early colonial societies were pretty simple. The common law could deal with most of it and for the small amount that remained, the local county court would just use a bit of equity here and there. Since the law courts were jury courts, people were generally happy that justice had been done even if it hadn’t been done in a way that would have passed muster in England.
If that failed, one could always appeal direct to the government, which was variously corporate, a Royally appointed Governor with or without a council, an elected assembly, or a combination of any or all of these. They could pass a bill that corrected the specific wrong.
For most of the 13 colonies, this is how it was until independence.
Some colonies in New England tried to establish Chancellery courts through their elected assemblies but these were universally struck down by the British Parliament as being beyond their legislative power. As a Royal Prerogative, the power to create them came from the King’s representative, the Governor, not the assembly.
Most Governors weren’t keen to exercise that power because they generally weren’t lawyers and they had a colony to run and didn’t have time for playing judge.
However, Philadelphia had an equity court from 1711 to 1732 which got very little use, hearing an average of 3 cases a year. It was unpopular with the assembly and the Governors.
New York had one from the early 18th century until independence. It was a political football that everybody liked to kick. Sometimes the assembly loved it, sometimes they hated it. Some Governors used it as part of their power politics against the assembly and some avoided it like the plague.
One must always bear in mind that common law courts are a conglomeration of tradition, politics, and sometimes reasoned, well-meaning, but not necessarily effective reform. No one in their right mind would sit down to design an efficient and fair justice system and come up with the common law courts system, past or present, existing in any common law jurisdiction. The advantage they have is that they adapt and evolve to their circumstances and more or less work most of the time for most of the people.