I have been having the EXACT SAME problem with my 13-month-old for the past few months. I was glad to read your post and read some of the responses from other moms. At least now I know he is totally normal!
Here are a couple things I've been trying. Sometimes they work, sometimes they don't!
I start saying, "Okay, let's change your diaper" about five minutes before I actually do it so he knows it's coming. Then I have him help me carry the wipes or get his diaper. I will also give him a wipe to hold on to and tell him to clean his face or something. He likes holding the wipe and pretending to clean up.
I also keep several objects around me that aren't normal toys. For some reason he loves to hold the nasal aspirator. As he gets fussy or tries to squirm I say, "Ooh..can you hold this for Mommy" and give him one of the distractors.
We also sing songs and all the stuff you described in your post. I guess you just have to do whatever works and try to do it as quickly as possible!!! Hopefully it will get better!
Good luck and let us know if you find any good tricks.
J.