Dissolving a polymer thicker than a paint layer tends to take very long. Nice if it works, but often discouraging. Ultrasound, like in a jewellery cleaner, often helps.
An alternative might be an acetylene+oxygen torch. The flame burns the plastic away quickly, more so with an excess of oxygen. It may melt the coins too, depending on how it's done, but the silver in the coins has often the same value as the coins.
Molten silver is just more difficult to sell than a silver coin. Such a coin is basically a stamping meant to guarantee a silver amount and purity.
If you try the torch:
- Blow pure oxygen once the plastic burns. The flame will be less hot than with acetylene.
- Do that outside! Nearly all burning plastics are toxic for real.
- Maybe air suffices.
Possibly a strong ultraviolet source would degrade the plastic at interesting pace, so it can be broken and brushed away, but I doubt it. Sunlight takes its time, especially if the plastic is black, thick, and resides between the coins. Some sources, like arc lamps, are stronger.
As most chemical methods get slower with the plastic thickness, it wouldn't be bad to first separate the coins with a chisel, hammer (and vice), leaving just thin plastic on them.
After that, just a barbecue fire followed by brushing may suffice. De-oxidising the tarnished silver is standard practice.
Hey chemists, would there be some compound (A bleach on steroids? Ozone? Percarbonate?) that oxidises the plastic away? That should be faster than a dissolution.
Or one that makes the plastic fragile (again an oxidizer?) so it can be hammered and brushed away?
Would a strong base be worth trying? Breaking the polymer chemically must be faster than dissolving it.