Replace the phrase 'sinusoid with time-period dependent on V1' with 'a sinusoidal function of V1' and you've nearly got it, except for an extra factor of two you shouldn't have, so the formula is:
Vout=V0*Sin(pi*Vin/V1)
Also the output is only specified over the Vin range 0 to 4*V1, which is important when you need to design a real-world implementation.
A simulation based on the mathematical transfer function is trivial (LTspice sim attached), but a real world circuit without using digital processing would be a *LOT* harder.
Lets suppose the input is a linear ramp ovwer the range 0 to 10V in time 4T. The desired output is a sinewave of period T.One possible approach would be to convert the linear ramp to a triangle wave then apply that to a sine shaper circuit. The conversion to a triangle wave would be made by inverting it, and appliying the original and the inverted voltage through various weighting resistors and clamps to a summing amplifier. The first section is just proportional to Vin, the next adds a component of -2*Vin above V1/2 to give the down slope needed between V1/2 and 3V1/2 etc. Then its just a very carefully biassed diode and resistor network that provides the sine shaping and an output buffer amp.
An alternative approach would involve a sinewave generator at a frequency a couple of orders of magnitude higher than the maximum input frequency, a circuit that generates a pulse on every other rising zero crossing, feeding a linear voltage controlled delay, that outputs a pulse to a sample and hold circuit to pick off the required output level from the sinewave.
I wouldn't build either on a bet, as an ADC addressing a lookup ROM feeding a DAC would be far more practical, and if the resolution and update rate is low enough, it could all be done in a small MCU.