Greetings! This question is an important one when it comes to investigating the history of the British Empire, and contrary to what some "pop-history" videos and sources might tell you, there was not a single formula" or "template" which the British applied to newly conquered regions. Firstly however, before we get onto that topic, we must discuss this bit of OP's question to dispel some misconceptions:

Did Britain have a template they would follow every time they conquered a new region to establish new governments and a system of law and enforcement?

Although the conquering of regions was the ultima ratio (last argument; final resort) of the British Empire, it was certainly an option which was the most effective. By conquering territory and forcing the surrender of a former leader/group in the region, the British were not only removing any potential rivals, but also gave themselves a better foundation with which to implement imperial rule of some sort. However, conquering a region was not the only way in which the British Empire expanded, at times their expansions were actually perfectly legal (minus the threat of using force), and you can find plenty of "cession treaties" in which a local power-holder peacefully transferred control of a territory to the British representatives on the ground. With that preamble then out of the way, let us turn out attention to the larger question.

Due to the very nature of imperial expansion, the British could never use a single formula/template for introducing imperial rule across the world. Depending on the territory and its sociopolitical, economic, and even environmental characteristics, the British had to be flexible and adaptive in their approach to implementing colonial policies. Consider for example, the case of South Africa, which I went further into here. Due to the fact that the British had strong rivals in the form of the Boer groups and native tribes like the Xhosa, they could not simply impose sweeping laws on the entirety of the region without getting into some form of conflict. Consider also the "jewel in the crown" of the Empire, the British Raj. Here the British East India Company represented the British Empire, and despite attempts in the late 1700s and early 1800s to bring the Company's territory on the Indian subcontinent under London's control, it took until 1857 with the outbreak of the Sepoy Mutiny for the British to firmly gain control of the region (more on that here with a great overview by u/conqueror_of_destiny) . Finally, consider the case of the 'white-settler' territories (Australia, Canada, New Zealand, and until 1776 the 13 Colonies in America). Here the British gave far more autonomy to the political assemblies, allowing all of them (excluding America once it gained independence) to have some form of Home Rule (that is, a parliamentary system similar in function to Westminster but technically independent of British control).

In all of these places and the other regions where the British had colonial interests, the same system could never be used because it simply would not work with the pre-existing laws, societal hierarchies, and power structures in those areas. Trying to give India home rule would have spelled disaster for the British on the subcontinent, whilst attempting to exercise direct control over any of the 'white-settler' colonies would have (and was in a fair few cases) been met with armed resistance by the local populace. It was up to the "men on the spot" to decide what form of rule (direct, indirect, or otherwise) was best for the to-be colony, and what specific institutions, patterns of rule, and law enforcement would best suit the local peoples.

Now, did these context-dependent choices tend to result in general trends in how the British Empire was ruled? Kind of. Historians nowadays often view three distinct categories of the empire and its government: the Dominions, the Indian Empire, and the Crown Colonies. The first of those systems is essentially what the 'white-settler' colonies evolved into (the name "dominion" was formally given to them in the Balfour Declaration of 1926). The Indian Empire was a massive bureaucratic beast, and its Viceroy was responsible for overseeing the day-to-day ruling of a territory encompassing millions of inhabitants. The final category, the crown colonies, was the largest "type" of imperial government, in which the British either directly implemented a colonial governor or colonial ministry to govern the territory, or they did so through the cooperation of local leaders.

That brings us onto a related question. If the government in London could not impose their will directly onto newly controlled territories or colones to-be, how important were the governors, governor-generals, and Viceroys in ruling the Empire? More specifically, how much autonomy did they have from "imperial command" back in Britain? We shall answer that question in the next bit of the response.

Part 1 of 2