London Screenwriters' Festival
Deadline: June 29th
The Bitch Pack
Regular Deadline: July 5th Final Deadline: July 12th
Early Deadline: June 22nd Final Deadline: July 27th
Early: July 1st Regular: October 15th Final: November 15th
As I was preparing to put this post together, I came across a comprehensive spreadsheet of competitions put together by Kulvinder Gill, it includes the above and many more. You can view the list here.
Finally, the BBC Writersroom keep a current list of writing opportunities on their website.
If you know of any other current or upcoming competitions or opportunities, please feel free to list it below.