The following sites have online games for children.  Most of the sites are character-based.  Some are better than others; well, presumably they save the better games for the CD-ROMs, which earn them actual money.

CBeebies/BBC
    Small online games, coloring pages, e-cards and activities.  There are activities for most of the shows on CBeebies.

Bob the Builder

Fireman Sam

Fun With Spot

Here come the Rubbadubbers, splish, splash, splish, splosh!

Nick Jr. Parents--Games
NickJr.co.uk - Play Home
    Both are Nick Jr, but each site has things that the other doesn't.  Adds a few more games and coloring pages to the repertoire.  I do find both sites a bit annoying to navigate.

Postman Pat - Official Site
    Postman Pat - Greendale Rocket Game
    Postman Pat - Julian's Lunchbag Game
    These two games are both on the Postman Pat website; due to the way the site is set up, I've found it easier to keep the separate links handy.

The Ragdoll

Ragdoll Presents - Brum - the super-brave hero

Thomas and Friends