Just wanted to share my experience of crossing the Yallama border (south of Tashkent) into Kazakhstan. This border apparently still is the only one allowed for foreigners coming from / going to Taskent. Individual travellers are still turned away at the Chernyaevka border (though I met travellers that crossed the Chernyaevka border while taking a direct bus or train to/ from Almaty but they all reported long queues and sometimes even angry crowds).
The Yallama border crossing was completely hassle-free and took me just an hour. Officers on both sides were extremely friendly and let me jump all queues. No bribes just big smiles.
I took a shared taxi from Sobir Rahminov metro station in Tashkent (20$ for the car) which took 45min. On the Kazakh side there are minibuses going to Shymkent for 15$ per person (3 to 4h).
