Which should be done first, lip lift or face lift? I want a different surgeon for each procedure?

TO specifically answer your question, facelifts typically  do not effect the same area that a lip lift does, hence, you can do the procedures in any order that you like.  

I usually perform both of these procedures at the same surgery, as I have quite a bit of experience performing both of these procedures.  If you decide to do the lip lift separately, the procedure itself takes an hour or so to perform, usually being done under oral sedation and local anesthesia, with a 7 day recovery period before the sutures get removed.

 I recommend that you find a surgeon you trust whose results you admire.  Then, review your options with them.  In general, I only would recommend a lip lift in a very select group of patients.  The order is not necessarily important but I would review in detail the recommendations.  Good luck.