I am tempted to say these boycotts are (as of yet) having no effect at all.
Having bought the idea up a couple of times, I believe the majority of academics (even in computing subjects) don't know they "should" be boycotting, and most have no intention of joining in when it is discussed with them.
I wonder if most of the noise you hear is from people attached to the 'free' journals, who obviously have an interest in getting people away from Elsevier.
Editors of Elsevier journals will send papers off for review, and get a growing number of "Sorry, I don't review papers for Elsevier journals." If you tell people that you are an editor for a nonprofit journal, everyone will agree that's a good thing, but telling people you edit for Elsevier will not have the same effect. Editorial jobs will become a drag. Morale will drop. People will quit. Journals will have to recruit lesser-known editors (and no one wants to submit good papers to journals with a bunch of B-list editors).
Meanwhile, people like Tim Gowers -- who is spearheading this movement, and who is also one of the most prolific mathematicians in the world -- will find other homes for their work. Wherever Gowers, Terry Tao, etc. publish, junior scientists would be very happy to follow. That is how the prestigious journals got that way in the first place.
"The company has been criticised not only by advocates of a switch to the open-access publication model, but also by universities whose library budgets make it difficult for them to afford current journal prices. For example, a resolution by Stanford University's senate singled out Elsevier as an example of a publisher of journals which might be "disproportionately expensive compared to their educational and research value" and which librarians should consider dropping, and encouraged its faculty "not to contribute articles or editorial or review efforts to publishers and journals that engage in exploitive or exorbitant pricing".[6] Similar guidelines and criticism of Elsevier's pricing policies have been passed by the University of California, Harvard University and Duke University."
Elsevier responding to pressure is not unprecedented; doctors protesting to the Lancet ultimately forced the parent company to sell off a defence trade show that was part of their exhibitions business (although divesting divisions of their trade businesses is something of a hobby for Reed Elsevier anyway). That had a difference though: they were pressurising Elsevier to take specific courses of action.
In this case, it appears to be more of a "we won't work for you because we don't respect your ethics" stance. Which is fine, but they'll have to continue reading and citing articles in Elsevier journals in their work to be respected as academics.
Ultimately, the value of copyrights giving Elsevier exclusive control over access to thousands of existing "must cite" articles in searchable format means it would continue to be a profitable entity if it stopped publishing new material of comparable quality or any new material at all. It's a drop in profits I'm sure the Elsevier board would rather avoid, but the boycott doesn't seem to be offering them any more attractive alternatives.
And are the other publishers looking at their business models so they can become less bad than they are now?