Today The European Aviation Safety Agency (EASA) has suspended Turkmenistan Airlines flights to and from the EU pending confirmation that it meets international air safety standards. This means that Turkmenistan Airlines flights between the UK (London Heathrow and Birmingham) and Turkmenistan (Ashgabat), do not have permission to travel to and from the UK. Affected passengers are advised to contact Turkmenistan Airlines to seek advice.
More information for affected UK passengers will be available on the Civil Aviation Authority (CAA) website.
https://www.gov.uk/foreign-travel-advice/turkmenistan

Update 07/02/2019:-

Wamos Air are currently operating TUA flights into the UK, and so far EC-MJS A332 has already operated to and from Ashkhabad from BHX.

Current BHX schedule:-

 

1200/1350 T5 425/426 752 from/to Ashkhabad operates on Friday
 
 1515/1715 T5 429/430 752 from/to Ashkhabad  operates on Sunday

1740/1920 T5 417/418 752 from/to Ashkhabad   operates on Monday

 1940/2120 T5 419/420 752 from/to Ashkhabad  operates on Tuesday
 
 2120/2250 T5 421/422 752 from/to Ashkhabad  operates on Wednesday