There are lots of islabikes on ebay within your budget. Then you can sell it on when it's out-grown.
Also check out race bmx bikes. They're £300+ new but there are loads of both 20" and 24" alloy framed bikes around secondhand, again there are plenty on ebay. They'll weigh and break less than the jump-bike-shaped-object from halfords.
There are lots of islabikes on ebay within your budget. Then you can sell it on when it's out-grown.
Also check out race bmx bikes. They're £300+ new but there are loads of both 20" and 24" alloy framed bikes around secondhand, again there are plenty on ebay. They'll weigh and break less than the jump-bike-shaped-object from halfords.