I don't know where you found information that winter tyres are required by law, but that's not true though.
In few European countries winter tyres are strongly advised or required in mountain regions - but not in Poland.
Other thing is, that if you're going to Zakopane region in winter time, driving on snowy roads without winter tyres can be very difficult and dangerous. So if you can, try to get a set of good winter tyres. But it's just advice. They're not legal requirement.
make sure you have esp/abs etc as it makes for better snow driving,
To be honest, where abs/esp can help greatly on wet surface, on snow they are not very good thing.
On snow most likely you could stop the car without abs faster that one with abs.
ESP can do different upredictable things on snow.
Better prepare yourself to handle car on snow, f.e by training recovering from skids, and not trust your car electronics too much.
Tyre chains are generally banned on the road between zakapane and krakow and aren't really necessary in all honesty. If you hire a car in PL or ze fazzerland it will come fitted with suitable tyres so don't worry about that.
Tyre chains are banned on roads without snow.
Whenever there is snow on the road you can use them.
But to be honest, most roads you can easily travel with winter tyres, and you won't need chains. Maybe some few really steep local roads might get some use from chains, but i don't thing you'll need them.
Driving in Poland is very differerent to in the UK and Germany due to the fact that all other roadusers are in fact incompetent egotistical idiots who overtake in stupid places and have no regard for other motorists. Theres other threads about this on here.
I know there is many drivers like this, but don't say everybody behaves this way.
I'd say maybe 20% of drivers drive dangerously or very dangerously. Everyone else is fine.
Make sure you have the full compliment of totally unecessary equipment in your car including warning triangle, first aid kit, spare bulbs, Fire extinguisher and those stupid high visability vests.
That's not true.
Only legal requirement in Poland is warning triangle and fire extingusher. You don't need to have first aid kid, spare bulbs or high visibility vest.
Moreover it's only required from cars registered in Poland. According to Vienna Convention, in international traffic, cars are only expected to have equipment required in country of registration.
So you, as a driver from UK don't need to have anything from this list.
If Police will try to give you penalty for lack of one of there, just show them the part from Vienna convention of international traffic.
You can print the act before travel.