To install channel updates without installing a system update you will need a homebrew application designed to update specific Wii Channels while avoiding a system update. The two I have tried are WiiSCU and Pimp My Wii, either of which can be used to update channels. To install either one just extract the application files to your Wii SD card's "apps" folder.
With WiiSCU you will be shown a list of individual updates for your channels, plus some other updates. WiiSCU has a simple interface but doesn't tell you what non-channel-specific updates need to be installed for specific channels to work. However, if you just go ahead and install all available updates there's a reasonable chance your channels will update properly. (When you select "IOS" updates you are given a choice of "-truchet" or "+truchet"; how to decide which one to choose is discussed here - it appears that most people will want "-truchet.")
Pimp My Wii is designed to simply go through your Wii and update everything that can be updated. It has successfully updated channels after WiiSCU failed to, but I admit it's a little unnerving to simply let an application do so much at once. It is possible to choose specific items you want, although the interface is a little confusing. You need to page through each possible update, choosing the ones you want, and then eventually you'll reach a point where it will update them all. In spite of the stupid name, I've had good luck with this one.