Franchising is not really a good idea because it will require complex licensing and different paper requirements I guess.

It can take from three weeks to three months in processing and getting PSW

check this out
http://www.immigrationboards.com/viewtopic.php?t=58184

Various people have given their time frame.

when you enter Uk use UK passport and while entering azerbaijan, you are supposed to show them Azerbaijan passport. The airline staff checks if all the required paper work is valid and authentic, ther wont be any issues dont worry:)