Any idea which ones they are? We stock Popolini, they do various nappies, the label should say if you can read it but here's a quick low down:
Popolini One Size - looks like a Mother-ease One Size, has a popper in booster to add absorbency. To use on newborns, remove the booster, fold the front down and out, and fasten the tabs to the poppers where the booster was attached.
Popolini Two Size - I don't think this is your nappy, this one is similar to a Mother-ease Sandy's, fastens at the hips with poppers and comes in 2 main sizes.
Popolini Ultrafit - this is a birth to potty nappy, has poppers down the front to reduce the rise and has hook & loop fastening.
Your wraps could be Vento wraps, are they popper fastening, with poppers at the hip and the leg? They work very well if so, good products, and actually we've found the Popolini nappies to be robust and reliable. You can add a fleece liner for dryness if you prefer, you can make your own or many nappy sellers stock them.