Final answer:
The preferred architectural style for British government buildings and elite private estates in the late 19th century was the Gothic Revival.
Step-by-step explanation:
The architectural style that borrowed techniques from Hindu and Muslim traditions and became the preferred style for British government buildings and elite private estates by the late 19th century was the Gothic Revival. This style sought to revive medieval forms and was associated with traditional sensibilities such as conservatism and the monarchy. It is characterized by stone and brick structures, heavy decoration, pointed arches, steep gables, and large windows.