Gap: just like in the States. Casual wear for kids with frequent discounts and sales. There are also numerous locations around London. www.thegap.co.uk
H&M: like in the States, but much nicer shops. Casual wear with a reasonable price tag. They also have frequent sales. http://www.hm.com/
Next: Similar to H&M. Good quality clothing and stylish. Also carry party wear items. However, sizes for children over 6 are often limited http://www.next.co.uk/
Zara: like in the states. Better quality than Next or H&M, but pricier. However, not true to size. Clothes often fit smaller than labeled.
Mothercare is known for its good quality, affordable children's wear and accessories.
John Lewis (Peter Jones): carries good quality clothing at competitive prices. They also have free delivery if spend over £30 online. www.johnlewis.com/
M&S: good quality and reasonably priced. www.marksandspencer.com/child
Harrods: described as "the most famous department store in the world", carries top designer labeled children's clothing
Selfridges: trendy and carries top designer labels. http://www.selfridges.com/
Fenwick: good quality, similar to John Lewis. http://www.fenwick.co.uk/
Debenhams: affordable clothing http://www.debenhams.com/
House of Frasier: carries designer labels. Tries to be overly trendy, but doesn't pull it off like Harvey Nichols. http://www.houseoffraser.co.uk/
Primark : cheap, but overly crowed and not great quality. Make sure to shop early and don't think about returns since lines can be extremely long.
ASDA: carries the George line as in the States.
Costco: just like in the States. Can find some great deals, especially on outerwear. Need to have a car to visit.
Amazon: when you can't find it or don't have the time, this is where to go. www.amazon.co.uk
Lands' end http://www.landsend.co.uk/