According to this Wikipedia article
, steam engines were first used in an attempt to create a steamboat in the 1680s, and the concept of the locomotive dates to the same period, although neither effort was successful in establishing the steam engine for those uses. This article dates the practical application of steam engines for pumping water out of mines at 1712. It credits James Watt with devising a more efficient and practical steam engine which was used in industry in England in 1769. This means that the use of steam engines in weaving mills probably dates to the 1770s.
The next development was the reciprocating steam engine (earlier engines used a "rocker beam" with the power translated through gear arrays), which is basically a steam-powered piston engine. Those engines were the first practical engines which could be used in steam locomotives and steamboats. There were many refinements of the reciprocating and balance beam (rocker arm) engines, but the next truly significant development in steam engines was the steam turbine. The steam turbine used fewer moving parts, produced far more energy (by orders of magnitude) and produced far less vibration (vibrations shortened the life of the engine by generating wear in the moving parts). The steam turbine, dating from the late 19th century, was perfected on a large scale only at the beginning of the 20th century. At the time when it became practical to build large steam turbine power plants, Winston Churchill was the First Lord of the Admiralty (England), and Admiral Jackie Fisher was the First Sea Lord. Between them, they decided to convert the Royal Navy to steam turbines (this technology is still in wide-spread use), which gave them more power so that larger, faster, better armed and better armored warships could be built. But steam turbines need very high temperatures, and burning coal to heat the water is no longer practical. Therefore, the steam turbine required oil-fired boilers, and the middle east suddenly became crucially important, as a source of petroleum. BP was not originally British Petroleum, it was the British-Persian Oil Company, and lead the English into Persia (modern Iran) in search of petroleum. The United States was not initially involved, because the United States is a petroleum producing nation, and therefore could convert and build a navy using steam turbines without becoming involved in overseas ventures to find petroleum, which is what Churchill and company were doing.
There are probably only three ways that your ancestor could have operated a sawmill. The oldest traditional method was the sawyer pit. A pit is dug in the ground, and two men use a very long (8' to 10') saw to cut planks from a log. One is the "top sawyer" (the guy on the ground above the pit, top sawyer was long an expression in English to mean the man with the good job), and the bottom sawyer, who stood in the pit, each of them pulling and pushing the saw in turn to produce the plank. This is not
likely to have been the method in the saw mill you describe, because this method was not described as being a mill.
Therefore, there were likely only two other possibilities. One would have been a watermill, where a large paddle wheel would be suspended over water to translate the energy of the water's flow into lateral force, which through a gear array could be used to operate a saw. This was a common method for centuries, and was long used in North America because the many rivers and streams meant that water power was plentiful and cheap. You might say it was free, but stream bank or river front property was very expensive precisely because it was so valuable to water mill based industry, and the mill itself had to be maintained, a constant expense since the drive shafts, gears and pulleys were usually made of wood, and the drive was usually translated from the drive shaft from the water wheel by use of a very heavy belt made of canvas. All those parts were subject to rapid wear, and that was increased by vibration in the mechanism.
The final method would have been to use a steam engine to produce the energy, although the rest of the equipment would have been very much like what was used with water power. For a steam engine to have been practical in the 1880s, a time when steam turbines were not being used in industry, there would have had to have been a source of fuel, wood or coal, to heat the boiler which is the heart of the steam engine. Tennessee in the 1880s would have been able to provide that, but it would have been relatively expensive in the 1880s, because it was not a coal-producing area, and he would have had to have used wood cut locally. Wood cut very far away or coal would have been expensive because of transportation costs.
Therefore, i suspect the sawmill would probably have been water-powered, as that would have been the cheapest, and most practical way to get the power necessary to operate the mill. In Tennessee, the Tennessee River and its tributaries could have provided the source of power for your ancestor's sawmill. As i say, a water mill would have been the most likely source of power to operate the sawmill.