Is it OK for a child's bike to have a radially-laced rear wheel?
For a kids bike where the rider weight is under 25 kilograms / 50 pounds a radial spoked wheel is fine.  

I've been trying to clean up and true an old 12" steel wheel with radial spokes and I'm truly astonished how strong it is.  The spokes themselves are identical to normal bike spokes, just the length that is different.  The cross sectional area is the same.

The rim is normally the same profile as a bigger wheel too, just with more curve on it.  This also contributes to the smaller wheel's overall strength.

I'd say a 16" would be fine, but a 20" and above should be properly laced.
If the rider is... over 25 kilos then perhaps a cheap wheel is inappropriate.
The rider will probably outgrow the bike frame in height before they get too heavy for it.