I've still got XP, so I'm not at all familiar with Vista conflicts, unfortunately. However, I do have some suggestions that may help.

If there's three computers trying to access the net at the same time, that will slow it a lot. Especially if trying to load high bandwidth sites, such as profile sites or video sharing sites.
If you have a go at accessing the net with all other network devices turned off, or disconnected from the same network and find you're back to normal speed, then that's probably the cause.
I doubt this is the case though because your other computer had no problems.
There could be a conflict between Vista and the modem. Your guess is better than mine on that.
It could be Avast because a search on Google shows a lot of people are having the same symptoms:
http://www.google.com.au/search?q=avast ... =firefox-aIf other programs on your computer are noticeably slower to load with your antivirus software running, then it's very likely that's the cause. To test this you'll have to disconnect from the internet then see how long a program takes to load up with the antivirus software running, then again after disabling the antivirus software. *If you have internet security software too, then you'll also have to test with that. Internet security suites are more known for slowing internet connections than antivirus on its own.
There's a bit about a McAfee conflict with Vista on this page of Microsoft's website:
http://social.msdn.microsoft.com/Forums ... 0f85b85be/From about 2/5 down the page:
"Finally cracked it (after switching off IPV6); turns out culprit (for me) was McAfee
Uninstalled and now she flies 
Hope this helps others"I have no idea what "IPV6" is.
If it turns out to be a conflict between Vista and your antivirus, it's probably best to see if there's a fix first before trying other antivirus programs because every time we uninstall things such as antivirus, they tend to leave behind bits of program in the Windows registry. One of these is because they add date stamps to show when the program was installed. They don't want people to keep reinstalling the program after a free trial has run out. Back in the old days of Windows 95 some people did that with several programs.
I hope I've been helpful!