Thanks to all for your help. I climbed up on the roof this weekend. Checked the first trap and it looked clear, as far as I could see. Checked the second - and there was the problem. A half-decomposed squirrel was stuck in there with many bugs mixed in for good measure. A pretty disgusting sight. After a while, I was able to pull it out of there. The toilet is much better now, obviously. I think it's still a big clogged from some fur, etc.
Will try the hose attachment this weekend to flush the last bit out. Thanks to all for your advice! Next, I'll put on some kind of screen over the openings, as I don't want to go 'squirrel fishing' again!