This Play Submissions Helper, a calendar-style blog, is focused on competitions in the USA, but a number of these are nevertheless open to London playwrights. Of particular note: the page featuring competitions with cash prizes.
Source: UK Playwrights Facebook Group.
If you’re feeling in need of inspiration, check out this BBC Radio 4 interview with Abi Morgan for FrontRow (Jan 2012, length). In it, Morgan discusses the differences between writing for theatre, television and film, her writing process, and how she juggles so many impressive projects. Aspirational and entertaining listening – all in the first 12 minutes.