yes it will be quite cold there, but i would suggest don't go overboard. you can buy the necessities such as woollens and warmers a few weeks before the trip, as children tend to outgrow clothes very fast. if you buy it now it may not be as good fitting as buying it then.
go for warm leggings that your child can pair with shorts or with tops. thick socks, comfortable closed shoes, a pair of gloves just in case, warm caps that can be pulled over the ears, thick sweaters and jackets.