I'd suggest people try using Firefox (www.getfirefox.com) and then in the add-ons you can browse for an advert blocker.
I personally don't allow Flash to run on all my PCs. Only one has it installed (and there's an add-on for Firefox which blocks Flash from running, unless you click on it - just displays a letter "F" to show where something is blocked).
I have today been using a friend's PC and finally (after weeks of seeing complaints) seen the annoying advert. Can understand that MN would not be in a very clear position to turn down ads, but perhaps MNHQ can give CBeebies "feedback" in the strongest terms that many users of the website have complained because they find this "animated" advert far more annoying than any other...
At least give some feedback that allows CBeebies to know it was seen but hated by many. Other websites have sometimes acted to remove such "animated" ads, when complaints are received, but the last time I saw a response from MNHQ was that it was "cute" and "only on for a month".
Well, however "cute", if it annoys readers, they will act to block not just that advert, but ALL adverts (eg placing ads1.mumsnet.com in a "blocked sites" list, so nothing from there will display, if they are tech savvy).