It used to be easy to walk into a bank in Uzbekistan and walk out with a personal account. They'd ask for passport, proof of address in Uzbekistan (bring hotel reservation), and some basic information about yourself. You could do it at practically any branch.
Nowadays, like everywhere, it's become a lot harder but AFAIK, you can still do it with most banks — for example KDB, Asia Alliance, Kapitalbank, Aloqa, Ipak Yuli, and a few more. But you need to go to one of the big branches.
On your first day, go get a PINFL.
PINFL for foreigners Takes 1–3 days. Once you have that, account opening will be much easier than without PINFL.
Company accounts are harder.