Not that I know of. Various experimental self-loading rifles were developed and tested at different times in the first half of the Century, but none was used in WW2. The combination of the Lee Enfield, Bren and Sten was obviously felt to cover the needs of portable small arms during the conflict.
As a result of WW2 experience, the UK did develop a selective fire rifle as part of an integrated package of weapons and multi-purpose cartridge, as described in 'Assault Rifle: the Development of the Modern Military Rifle and its Ammunition' by Max Popenker and myself:
'A weapon which very nearly did see service was the British EM-2 rifle developed in the late 1940s. Unlike the AK 47, which continued to be supplemented by the full-power 7.62 x 54R Nagant cartridge in MGs and sniper rifles, this was a carefully-judged attempt to produce a weapon which could replace both the 9 mm SMG and the full-power .303 rifle in one package, by combining a new .280 (7 x 43) intermediate cartridge with a compact 'bullpup' layout. A GPMG based on the Bren LMG mechanism but with belt feed, the TADEN, was also developed to use this round and replace both the Bren and probably the Vickers MG. It appears to have been very successful and other NATO countries (particularly Canada and Belgium) were very interested in the concept. The UK even formally adopted the EM-2 in 1951.'
Tony Williams Military gun and ammunition website:
http://www.quarry.nildram.co.uk Military gun and ammunition discussion forum: