Cash rules. You will get a better deal, you can haggle to get a lower price. Atleast in UK, but not sure how it works in Poland.
rent it on Airbnb
This might be a simple option, but income generated on Airbnb will be classified as company income and will be taxable.
work remotely from there.
This will be a complicated operation and my advise ( and if you have an accountant ) will be to stay away from it. This will be classified as benefits under UK tax regulations and you will need to pay class 1 NIC. So not worth it.
And getting a mortgage for a LTD company in UK, you will only get a Buy you Let mortgage whose purpose is primarily to recycle your rising equity (ideal condition) in the house by remortgaging. In UK because of restricted house supply this is ideal, not sure if it the case in Poland. Secondly you might ask Polish lenders too. I think it will be simpler if you do it through a Polish entity rather than an UK entity. But doing this is going to complicate your LTD company tax matters, so think before you leap into it.