Research
The effect of advertising on children is an emerging area of research. As with most psychological studies the results are a little less clear cut than those in the field of physics or chemistry. However those studies that have been carried out should give most parents cause for concern. Here are a selection:

Watching, Wanting and Wellbeing National Consumer Council 2007

Child Wellbeing in the UK, Spain and Sweden UNICEF report 2007 suggesting that children from the least well off families are most adversely affected by advertising.

The Impact of the Commercial World on Children's Wellbeing: Report of an Independent Assessment A report to the DCMS 2009.

The WHO produced report A set of recommendations on the marketing of foods and non alcoholic beverages to children is forming the basis of action around the world and reflects the impact of advertising on children’s food choices. It’s not yet being followed in the UK.

Meanwhile here is an ASA ruling on a complaint relating to an advergame. The ASA refused to act without evidence of harm. Even in cases of physical harm, as with tobacco, change predicated on this sort of evidence can take decades.

How other countries do it:
Greece:
Greek legislation prohibits the broadcast of toy advertisements aiming at children from 7 a.m. to 10 p.m.

Quebec:
248. Subject to what is provided in the regulations, no person may make use of commercial advertising directed at persons under thirteen years of age.

Sweden:
Radio and television act, Section 7. Commercial advertising in television broadcasts, Teletext and on-demand TV may not be designed to attract the attention of children less than 12 years of age.

Commercial advertising in television broadcasts and on‐demand TV may not appear immediately before or after a programme or part of a programme that is primarily directed to children below 12 years of age.

Norway:
Broadcasting Act Section 3-1 says:
“Advertisements may not be broadcast in connection with children's programmes, nor may advertisements be specifically directed at children” “Advertisements may not be broadcast 10 minutes directly before or after children’s programmes”