BA do want to fly London-Atyrau but several other European carriers want to fly to Atyrau as well, and there is no immediate prospect of this changing even despite the runway upgrade at Atyrau. Other works appear to be required before the airport can receive more traffic. It was BA mainline; and they were looking at Gatwick-Atyrau with a 757, but quite how this would work when the 757s are all based at LHR is something of a mystery.
The only recent development has been Astraeus re-starting its oil industry flights between Gatwick and Uralsk and then starting a new service between Gatwick and Aktau, further down on the Caspian.