How to find ICM talks?
Update: (Oct. 2018) For the first time, all ICM 2018 lectures (plenary, invited and special) as well as panels and special events are presented (by good-quality videos) on the ICM 2018 You tube channel.
Update:(Dec 2017) The ICM launched a new website. All previous ICM proceedings are available here. However, I cannot find the old page with access to individual papers and search options.
Just recently the International Mathematical Union (IMU) put online all the previous proceedings on the ICM's! Here: (update:broken). This webpage is based on joint work by R. Keith Dennis (Ithaca) and Ulf Rehmann (Bielefeld). So you can read for free all the articles (including Hilbert's famous problem paper; Martin Grötschel demonstrated it at the opening ceremony of ICM 2010). I don't know when the proceedings of ICM2010 will be added. UPDATE: the 2010 articles have now been added! Update The ICM2014 papers are available here (you can download each of 4 volumes; the IMU site does not contain these papers yet.)
As for the talks themselves, there is a page with all the ICM 2010 plenary talks here; links for videos from earlier ICM's (plenary talks and other events) can be found here.
Update (August, 6 2014) Many (46 for now) of the ICM 2014 proceedings contributions are already available on arXiv, via this search. (I got it from Peter Woit's blog.) Videos of lectures can be found here.
Update (May 2014): Starting 1992 there is also every four years the European Congress of Mathematics (ECM) that the European Mathematical Society (EMS) is running. The proceedings of the first three ECMs are now freely open. These volumes are available here.
(Digitising the proceedings of the first three ECMs, published by Birkhäuser was a task carried out by the EMS Electronic Publishing Committee.)
Starting with the 4ECM (2004), the Proceedings are published by the EMS Publishing House. The EMS decided to make them freely available online too. I expect that this will happen soon and I will keep you posted. Further update (June 2014) The ECM Proceedings are now available here! I was told that in a few months, the EMS will put also the 6ECM volume.
Years ago I received for free a CD-ROM from World Scientific containing a PDF file with some Fields medalists' talks at the ICM. It contains talks up to and including the 1994 ICM. I have just managed to find this CD and put the relevant PDF file (11Mb) here for anyone interested.
Edit In December 2020 received a cease-and-desist email from World Scientific asking me to remove that file from the server. The above link is therefore now broken, alas.
The IMU acts as sponsor of the ICMs and maintains a fair amount of information at http://www.mathunion.org/ including a list of all past venues with online links from 1998 on (as you've already encountered). For instance, the 1998 Berlin ICM proceedings are all freely available at Documenta Mathematica's site http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/ICM.html. But that's an optimal case.
I've heard in the past that there was some movement toward scanning all past proceedings volumes, but this might not be easy due to varying copyright ownership (and costs of the project). Many but not all libraries have those volumes. It's always a problem to find a publisher, commercial or nonprofit, to handle the immediate task of publication after a Congress. In recent years most invited talks have been available on the arXiv, for instance one just posted this week by Ivan Losev on finite W-algebras. But that's up to the initiative of the speakers.
Cautionary footnote: While the ICM can be quite a good experience for those attending, the published talks don't always give helpful overviews and are often highly technical. Better surveys might be found in many places, including the arXiv and the AMS Bulletin and Notices (free online). In any case, the posted Fields Medal awards and responses are a useful resource.