I have been using kweather 2.1.0 (with KDE 3.5.10) for some years.

Recently I noticed that this desktop application no longer calls up noaa.gov for weather information but now seems to call up to a colocation server (?) operated by Akamai Technologies.

Anyone know or can guess anything about this? Is it a possible security concern?

Have you even tried to inform yourself about Akamai? I bet you have used their services many times without even noticing it. Have a look here, especially at the customers section.

Of course I have studied available sources of information on Akamai, including the "encyclopedia" article you mention. My concerns have not been allayed by what I found. And you didn't address the point that kweather's behavior changed: it was calling up NOAA and suddenly it's calling up the colocation servers operated by Akamai.

Moving on, kweather is not important and it's starting to worry me, but apt won't let me uninstall kweather without uninstalling KDE itself, which is important. What's the best way of disabling kweather permanently?

Remove the executable bit from it. But you won't get away from Akamai.