NOTE This is a work in progress
Preamble
Right. I have this up and running on my US Wii and it's well worth the effort. In summary the benefits are.
1. Play Gamecube and Wii games from any region on your Wii
2. Access region specific content (e.g. Virtual Console and Wii Ware titles from other region's shops)
Possible cons are the usual stuff:
1. Slight risk of bricking your Wii
2. Might invalidate the warranty
3. Uncertainty of compatibility with future system updates
On balance I think the pros outweigh the cons, but then that's just me 
Pretty much everything you need to do this can be downloaded from this site.
http://hbc.hackmii.com/download/
What you will need
You will need a copy of Zelda Twilight Princess (Wii version) that has been played and saved at least once; an SD card of 2GB or less in size and means of accessing it from a PC (e.g. a card reader or failing that a digital camera with SD and USB interfaces) and an archive manager for extracting the files from the packages. On Windows I favour 7zip http://www.7-zip.org/ for Linux I just used the default archive manager that comes with my distro.
Right got all that together?...
Step 1 - Download the files we need
The first stage is to download the following packages from the link above:
1) The twighlight hack (http://hbc.hackmii.com/dist/twilight-hack-v0.1-beta1.zip)
2) The homebrew channel (http://hbc.hackmii.com/dist/the_homebrew_channel-beta_8.tar.gz
3) The apps package (http://hbc.hackmii.com/dist/hbc_apps.tar.gz)
I've included the links to the individual files above to make things easy, but it's worth referring back to the hackmii site to make sure you're using the latest packages The rest of these instructions are based on the beta_8 release of the homebrew channel. I recommend you check the readme file included in the homebrew channel archive if you are using a different version
Step 2 - Prepare the SD card
Now let's format the SD card as per the instructions in the homebrew channel readme, which says the SD card needs to be formatted with the FAT16 filesystem. This step may be unnecessary if your card is already formatted as FAT16. Obviously formatting the card will erase all of the data on it. If you've got files you want to keep then back them up first.
In Windows:
Pop the SD card in your card reader. Then,
Start menu -> Run... then type cmd into the run dialog. This will open up a terminal. The command you need to format the card is:
FORMAT (drive letter): /FS:FAT
replacing drive letter with the letter that the system has allocated to the SD card. For example if your SD card is F: the full command will be:
FORMAT F: /FS:FAT
In Linux:
Pop the SD card into the reader. If your distro mounts it automatically then great otherwise your going to have to mount it manually.
Open up a terminal and run the following command:
man mkdosfs
I can't be bothered to elaborate more than that at the moment. The man file should provide enough information to get you through.
Step 3 - Extracting the downloaded archives to the SD card
Now we need to extract these archives to the SD card:
1) Extract the 'private' directory from the twilight-hack-v0.1-beta1.zip archive to the root of your SD card
2)
Step 4 (optional) - Adding additional applications to the homebrew channel
I'm going to use the AnyRegion_Changer application here as an illustration of how to add applications to the Homebrew channel. The Any Region Changer application is especially useful as it lets you mess about with the region settings of the console. The primary use for this is to access another regions shop. For example I used this to download Gradius Rebirth from the Japanese Shopping channel on my US Wii.
Step 5 - Performing the Twilight Hack and installing the channel
The Twilight hack is a game save exploit using the Wii version of Zelda the Twilight Princess. Basically what you need to do is replace the Twilight princess save game file on the Wii system memory (remember I said you had to have played and saved the game at least once for the hack to work) with a special save game file that enables the running of unsigned code. The key thing here is that there are many different versions of the Twilight Princess each with their own corresponding hacked saved game file. The Twilight hack package you downloaded under Step 1 and extracted to the SD card should have every version of the hack included, so it's a simple case of selecting the right one to use.
To ascertain which hack you need take a look at your Twilight Princess disk. Near to the centre of the disk you should be able to make out some letters and numbers. These tell you which version of the game you have as follows:
***insert decoding advice from readme here***
Now that we know which hack to use go to the Wii system menu, data management and copy the correct save game from the SD card over to the Wii system memory, overwriting the existing Zelda save (Note this will erase your Zelda save file. If you want to keep your progress back it up first on to the SD card.
Now back to the home screen, select the disk channel and boot up the game as per normal. Select to continue game and pick the hacked save game file from the list of saves presented to you. Once the game has loaded and you are in control of Link move him backwards (pull back on the Nunchuck analogue stick). The screen should go black and a load of white writing appear. Don't panic this is normal. If you've put all your files in the right place on the SD card the hack should pick up the homebrew channel installer .elf file and run that. Follow any prompts that the homebrew channel installer gives you.
Concluding remarks
That's it you're done! Have a cookie as you've just made Yoshi cry.
Addendum
One potential show stopper, which is tripping up some poeple is the Homebrew channel can't load the applications from the SD card if you have a Gamecube memory unit inserted in the Wii. I stumbled across this by pure chance, but it's worth remembering if your apps aren't loading up.

Thanks so much for that Madbury, you're a legend.
I really need to sort my machine out asap and this this stuff installed while I'm at it, the drive's properly on it's last legs now.
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One interesting thing, switching the video from NTSC to PAL on a US Wii with the AnyRegion changer enables RGB output, which looks very nice indeed. This is on a launch US Wii too (literally around the 500th sold), which suggests all Wii consoles support RGB in hardware, and Nintendo curiously chose to enable/disable RGB or s-video depending on the region.
This does break some NTSC games, mainly early VC games but also some disk based stuff. I'm guessing they get confused by PAL/RGB mode as they don't expect to support that (and are probably checking for 480i/480p before being told something unexpected).
From what Mads has said it seems like Gecko and Freeloader also switch video mode in this way, which disables s-video, so importers using s-video cables would be best switching between RGB and s-video if they often run a mix of PAL and NTSC games.
That's very interesting - I was dead certain it was the same video output chip they were using in the GC Component/D-Terminal Cable on the Wii board.
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I think it is, they are switching between s-video and RGB out from that in hardware somehow.
Question.
I'm aware that Scart can either do RGB or S-Video, by utilising common pins for different signals. Would it theoretically be possible to rig up a cable (with a switch say) that allowed you to flip flop between RGB and S-Video. It's going to be a ball ache to switch cables everytime I want to play a PAL game, which if my understanding is correct is the current situation?
It should be possible yeah.
Looking at the Wii A/V output:
http://www.gamesx.com/wiki/doku.php?id=av:wii_multi_av_pinout
and scart:
http://www.gamesx.com/wiki/doku.php?id=av:scart_connector
http://en.wikipedia.org/wiki/Scart
http://utopia.knoware.nl/users/eprebel/SoundAndVision/Engineering/SCART.html
You'd want to start with an RGB scart cable, as that has all the pins you need connected on the Wii A/V plug.
To convert to s-video scart you would need to:
- connect Wii A/V pin 9 (s-video Chroma, Green in RGB mode) to scart pin 15 (Red in RGB mode)
- connect Wii A/V pin 7 (s-video Luma, Red in RGB mode) to scart pin 20 (composite video in RGB mode)
- disconnect scart pin 16, or reduce voltage to 0 to 0.4v, to disable RGB mode on the TV (may not be needed depending on the TV)
I think thats it. You could implement this with a 4 pole switch, which is a bit awkward. Do you have a scart socket that accepts both RGB and s-video though?
Possibly a better option is just to tap audio and s-video from the scart plug, and have both connected at once switching between them on the TV (which you would need to do anyway).
Possibly a better option is just to tap audio and s-video from the scart plug, and have both connected at once switching between them on the TV (which you would need to do anyway).
I would say that's the best option aswell. Just wire up your S-Video to the appropriate pins and screw the clasp over the both cables, that way you can have your SCART plugged into AV1 and your S-Video in AV2.
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I believe AV1 on my tele supports RGB and S-Vid, but would need to double check that.
I like the idea of popping out a few wires from the scart block and wiring up a couple of phono connectors and Min din for the S-Vid. I've pretty much got the bits already to do that. All I need is a Wii RGB scart that I can butcher and I'm in business. The only slight downside is that I run two switch boxes on either side of my tele at the moment (one for RGB and one for S-Vid), so I'd need to rationalise those and make sure the S-Vid and Phono cables are long enough to reach from one side to the other, no biggy I suppose
Incidentally Maplins are doing a nice Fully wired manual Scart switcher at the moment that can accept 5 Scart sources and switch to a single output. £25 seemed reasonable. The Argos switchers are great, but I'm tired of switching cables over and didn't fancy daisy chaining them.