In the UK, the lower end of value ranges of nappies were generally plastic-backed in at least the late 2000s or so. My sister had some like that called Tesco Value I think. I would love to find a picture, but a Google search only gives me a few Mumsnet (a parenting forum, very middle-class) posts describing the nappies, some calling them 'manky' for their plastic-backing.
In the USA, Huggies kicked off the trend by shaking off the plastic about 1996 or so, the last major brand to do plastic backing was about 2001. These Luvs are from 2000:
I don't like baby diapers that much, but I don't mind those.