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How would a mechanism look that allows a person to e.g. turn a crown-sized wheel (from a watch) by any amount (e.g. 180 degrees) resulting in a turntable (from train yards) turning the same amount (e.g. also 180 degrees) ?

The mechanical amplifiers I found on Wikipedia all used inherent physics to allow a single input to transform into a single output, but in this case that would not be possible, since an additional source of energy would be required.

In electronic systems one might use a transistor (e.g. a MOSFET) where one can tap an available source of energy (e.g. a watermill) to amplify the input.

Size, complexity, cost and latency are not of concern, the only requirement is that no more advanced electrics are used (late Victorian at the latest).

You can go the hydraulic way or the electrotechnical way. This answer concentrates on the electrotechnical option.

The standard electrotechnical (as opposed to electronic) amplification device is called an amplidyne. In real history, amplidynes were invented during WW2 by the famous Ernst Alexanderson, the inventor of the Alexanderson alternator, an early high-power (and purely electrotechnical!) radio modulator.

An amplidyne is basically an electric motor (which provides the power) driving a DC electric generator (which receives the input signal as excitation current and produces the output signal); amplification factors up to 10,000:1 are possible. The massive drawback of amplidynes is that they work only with low frequency input signals; but I think that this is not a problem for the stated application.

Aplidynes can be purely electrotechnical devices, and there is nothing in them which is strictly necessary and wasn't available in the Victorian era; all that's needed is an engineer of Alexanderson's calibre. (Real-life semi-modern amplidynes used vacuum tubes for signal pre-amplification.)

So the solution I propose is to have the control wheel connected to a rotary encoder to convert mechanical position into an electric signal, which is then amplified by an amplidyne, with the amplified signal fed to an output selsyn (also known as a synchro) to actuate the turntable.

The Torque Amplifier is a device which does exactly this. It was invented in 1925 to allow manual control of heavy equipment, but does not require any technology not available in the Victorian era.

It is used in many devices including power steering, as well as the "ball and disk" integrator, used in mechanical analogue computers such as the brass brain of the Pershing nuclear missile.

In a nutshell, a small turning force applied to a spindle causes a cord or rope attached to the spindle to tighten against a moving drum. The drum then pulls on the rope by friction, amplifying the force. To make it work in both directions, two drums are required moving in opposite directions.

For more on the Torque Amplifier see Wikipedia:

• https://en.wikipedia.org/wiki/Torque_amplifier

For more on the MGM-31 Pershing the Wikipedia article is also good:

• https://en.wikipedia.org/wiki/MGM-31_Pershing

The trick here is counterweights (or just weights in this case).

A large weight goes from the thing to turn and passes over pulleys.

When you turn the small wheel you are not directly connected to the large wheel, instead you allow the weight to drop by a certain amount, pulling the wheel in the correct direction.

Turning it the other way would cause the weight to drop again pulling in the other direction (or you could use 2 weights, one for each direction).

Some other mechanism either manual or perhaps water or steam driven would winch the weights back up to the top when they are not being used to drive the wheel.