I run kids cricket at my club and a fair few of them turn up with size 3 Kashmir willow bats. there are 2 reasons for this
A. Sport direct bats seems to start at size 3
B. Kashmir willow is cheaper.
however, these bats tend to be quite heavy, particularly bottom heavy.
At this age they don't need a 'gun' of a bat, but something light and preferably something with a thin handle to match their small hands.
When I was looking for a size 2 for my daughter I went to Somerset county sports and there they had a good selection of kids bats. the Spartan and SS were particularly good as although they were Kashmir, they were relatively light and had thin handles. They were priced at £28.50 so I looked online to see if this was a good price, I couldn't find any comparables. At the time a second hand English willow size 2 bat came up on eBay so I bought that instead. It is a good bat, significantly lighter and responds well too.
I got a gun of a size 4 GM Duelist from the shop at the municipal tip. they always have a few bats there and they only cost a few quid.
For training I mainly use plastic bats for the kids in the 6-8 age range as they are significantly lighter. I got those from upfront cricket online for a tenner each posted. I am sure the sponsors on here do them too.