Don't think residency status is relevant.
AFAIK the only docs you need are passport and/or birth certificate. You use this to get a certificate of no impediment (ie, proof you're not married to someone else). Not sure where this needs to be posted - ie, will you need to post it in your local urząd miejski in Poland or in the Polish Embassy in London? - but need this you will.
Go to the Register office in which you want to marry. They will advise you. As long as you have not been married before (in that case you need divorce papers) you can marry anyone you choose.
You need a passport, birth certificate, (marriage and divorce papers - if you have a different name from the one you were born with.)
If you want to marry in Church go to the Church, but you will still need a registrar present and will need to sign the book.