Convolve a minimum phase wavelet with delta peak
If I understand the question properly. Consider the case when the delta is at time zero. In this case the convolution does not do anything to the other signal i.e. it starts at time y and the maximum peak will be at time z.

Now, if we delay the delta to time x, then the output of the convolution will also be delayed by x. So the signal will start at time x+y, and the maximum peak will be at time x+z.

If I haven't understood correctly, you may need to include some figures to more clearly explain the situation.