No - such things don't exist yet - banking laws in individual EU member states are different so any bank operating in the UK and Poland would have to run their operations as completely separate organisations.
Your best bet Is to send money between your Polish and UK banks using a money transfer broker which would (hopefully) lower the transfer fee and certainly convert currency at a much better rate than a bank or kantor.
I use World First and can recommend it. People also speak well of Transfer wise. There are others, but those are the best known ones.
Another tip is that if you're using banks in more than one country and sending money between them, choose very big banks. Smaller banks or ones that belong to other banks cost more when making international transfers because they have to use an intermediary bank who take a fee as well as your own bank's normal transfer fee. Bigger banks are "correspondent" with each other so transfers are a bit cheaper.
Also have a look on moneysavingexpert.com for the UK debit/credit cards that are best and worst for using abroad.
