There is an incredible amount of data out there from the analysis of the effects of water on brewing beer. Not that it is all applicable here, but it might make an interesting point of comparison.
The quality of water and the various minerals and ions in it can have a dramatic effect on the end product of a brew. When the water hits the malted grain and the enzymes get to work breaking complex carbs into various sugars, the ph is a very important factor for both conversion and flavor. When the sugary water is boiled with hops and other adjuncts, the minerals and ions start to play a bigger role as they effect how the oils and flavors of the hops integrate with the brew water. When the brew is fermenting, both the ph and minerals matter in the performance of the yeast.
This, of course, ignores about a zillion other factors in determining the taste and quality of the final beer (quality of ingredients, sterility of the system, temperature, etc.) Nonetheless, water is an important factor that brewer spend a lot of time dealing with.
For brewers who try to replicate the waters of other cities (generally well known brewing cities), they usually start by using a good filter to remove any chlorine and bacterial and organic matter. Then, using gypsum, salts, baking soda, acids, etc. they adjust the properties of the water to get it close to the levels of minerals and ions in the target city's water. This generally works quite well with the exception of biological agents in the target water which can't really be replicated (to my knowledge).
As to the effect on bagels: I would imagine that the effect of the boiling water on the final product would pale in comparison to the effects of the flour, yeast, water in the dough, and temperature/length of the rises. At the point where the bagel is boiled, I would venture a guess that most of the flavor has already been established and boiling would mostly effect the texture of the bagel. But I could certainly be underestimating its importance. I guess it depends on the water's ability to penetrate the dough and swap minerals and compounds.
This is a very interesting discussion.
Taft