Every UK bank is connected to civic registry. None UK bank will open account for you if you are not registered as UK legal resident in a civic registry. You are allowed to apply for registration just if you are a legal resident.
Just having a sublet appartment does not make you a legal resident of the UK.
You as citizen of
Slovakia have the same rights as anyone from each third country in the UK. Just etering UK with slovak passport does not make you a UK resident. You are just a tourist.
So the only way is to apply for
bank account as a foreigner with a foreign proof of address. But not as UK resident. The most of high street banks will be happy to open a saving account for you if you are willing to deposit at least 100k GBP.