Actually, Bizzy Lizzy had four wishes per day, if she wished a fifth time, then all the things she had wished for would disappear.
Usually her first wish was to make her Eskimo doll, Mo, come to life and "run about and play".
Bizzy Lizzy started life as cut-out animation inserts in Watch With Mother's Picture Book, before getting her own (marionette) series of 13 x 15minute episodes, also as part of Watch With Mother, in the late 1960's. The series was made in black and white, so didn't last long after the switch to colour transmissions.
Bizzy Lizzy lived just down the road from The Woodentops, as she and Mo visited Buttercup the cow in the last episode. The puppets of Teddy from Andy Pandy and a couple of Woodentops also appeared in Bizzy Lizzy playing other characters...
http://www.offthetelly.co.uk/childrens/watchwi thmother/part3.htm