Phase of a signal with multi delayed windowed (rectangular window) signal with different amplitudes
Assuming they all have the same $\omega$ and $\alpha$, meaning they are all following the same frequency ramp, then you can multiply by a ramp in the opposite direction of a starting guess while measuring the derivative of the phase of each carrier (a simple approach to see this would be to observe the result with an Short-Time Fourier Transform). Point is once you have found (and removed) the frequency slope, you will be left with distinct and constant frequencies, offset in frequency based on the time difference of their starting position and the frequency ramp rate (which also gives you information as you converge toward the final frequency ramp).



Since you do not care about the amplitude, then a metric that simply looks at the frequency components independent of amplitude would work well for this. 

For the frequency ramp correction, it would be simplest to do that by multiplying by a complex frequency ramp ($e^{-j(\omega t + \alpha /2 t^2)} $ ) rather than a cosine and then observe just the positive frequencies in the complex output (What you would observe from bins 0 to N/2 in the STFT).