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.
Best Answer
Update 2018.09.22. Go to the website icm2018.org, and use the menu icon (three horizontal bars) to get a menu list that has ICM videos on it. That takes you to a YouTube channel which has some of the plenary and invited lecture videos being uploaded. If there is a problem accessing or playing the videos please let me know, and I will relay the information. Marcelo Viana says in an email that they are uploading the videos, but did not say when they would be done. I checked it out and saw some of the plenary lecture titles uploaded. Enjoy! End Update 2018.09.22.
I will ask the organizers tomorrow. However, I suspect from other things that the plenary and invited lectures will be given lower priority than the current output, as much is being done to popularize mathematics for Brazil. My guess is that Mark Sapir is right, and that you will have to wait a few days for the lectures meant for specialists (non-public). As a second guess, I imagine they are preparing Portuguese translations, and that doing so within ten days will be an impressive feat.
** Update 2018.08.04 **
Marcelo Viana graciously offered a moment of his time; he says he hopes (with a pessimistic estimate) that the end of month will see the videos posted. They have been working hard to make this Congress as paperless as possible (which on the whole is a good move, I think), and one of the tasks remaining is to update and post (a version of) the database of abstracts for use by the Congress members (and eventually the public, I imagine).
Gerhard "Impressive To Me, At Least" Paseman, 2018.08.03.