If you have an R tailpiece, it sometimes helps to slightly bend the ball end of the string up in a slight curve to help it slot the notches in the tailpiece better.

Using a capo, masking tape and folding up the Ric polishing cloth and putting this cloth underneath the R tailpiece (or trapeze) are all good approaches for this process, in order to hold the strings securely and protecting the finish from accidental scratches.

I don't like to use tape, but I do use a variant of the Paul W (Jingle Jangle) method for the 12-string. It goes by pretty fast; I also bend the ball end of the string (it's a good idea for any slotted TP, or Bigsby, btw). I generally use a capo for the 12 string, to hold tension on the string and keep it from falling out of the slot when I wind it.

The last time I changed my 12-er with an R TP the process went by fairly quickly; it doesn't take that long; I think I did it in about 30 minutes or less start to finish, while chatting away with a guest. I think Paul W's record is 18 minutes, or slightly less.