Variant of discrete fourier transform that isolates phase delay?
Since this is a comparison of a template image, to a spatially, (and circularly) translated variant of it, then the power spectra of both will remain the same, but the phase spectra will be different. 

I believe in your example, the approach of a simple circular convolution between your image, and the template will be what you want. The result is also an image, and once you have that, you search for a peak. How much off in the vertical/horizontal direction that peak is from the a peak that you would have normally got had you circularly convolved the template with itself, will give you the spatial delay you have in both horizontal and vertical space respectively. 

You can implement that by taking the (same size as images) Fourier transform of your template image, (first one), and multiplying it with the Fourier transform of your image, (second one), and then inverse Fourier transforming the result back into the spatial domain. Once there, you can look for the peak as mentioned in the first paragraph.

This is assuming your images are the same size, as you stated.