can i get a temporary passport or something else really quick?
Maybe. You'll need to book an appointment through the passport service at your nearest office (google their address and phone number). They do a one-day service and you'll have to go in person (with the form ready filled in, get it at the post office) and some photographs (they may have a booth there). I had to do this once and it was quick. It does cost more. As I say, they have an appointment system and I'm not sure if they can deal with people who just turn up. If you explain when you phone that it's an emergency they may squeeze you in right away - I found them easy to deal with.
You say you're half Polish. If you have a Polish ID card, I think you can fly on that, however it would be a very good idea to check with the airline before booking. My ex flew from UK to Poland on an old style Polish ID card that had just expired and had no problems, however they may well have tightened up security a bit now.