How to estimate ALL time-delays across MULTIPLE signals?
There are papers describing a method called multichannel cross-correlation, that recursively estimates time delay of each signal. 

However, If I have not missunderstood you problem you can also select one of your signal to be your reference signal and then repeatedly calculate cross-correlation against that compared to the other signals. that way you can align every signal to your reference point.

EDIT: based on your latest comments:


  So far I've only had a chance to look at the original paper linked (microphones in a sound room). It's very interesting, however it seems to require that you know something about the geometry of the microphones. I had been hoping to find a generalized, system-agnostic method to do multi-channel cross-correlation. Something with which you can just input a set of arbitrary signals or patterns and it outputs the "best fit" delay vector.


What it appears to me that you need then is  blind system identification methods which I can link you some examples. Common methods for this task utilizes cross correlation error minimization but there are some limitations based on your model selection. 

Under-Modelled Blind system identification - cross correlation based  

Blind system Identification for acoustic source localization - Eigenvalue decomposition method

Blind system identification using sparse leaning method - LMS adaptive algorithm based on cross correlation formulation