Finally, I've figured this out, after many months of being unable to make any changes to my templates.
The folders /myforuminstall/templates/ and /myforuminstall/templates/compile had permissions set to 755, and the compile folder was full of cached pages. Since my UBBT couldn't write to the /compile folder, it kept using the old cached versions of the pages even after I uploaded or modified the .tpl files.
I set permissions on both to 777, deleted all the cached files, and now all the template changes I made months ago are suddenly there. WHEW!