the 303 Brit was determined to be obsolete as a military cartridge and a dead end development.
Was it declared obsolete though?
What I've read, is that there were many complaints stemming from the Boer War - where it was found that there was a marked difference in effectiveness between the 7x57mm Mauser round and the various round nose variants of the .303 British cartridge (with the Mk. III, IV, and V hollow point rounds removed from service due to the 1899 Hague Convention - seems they were a little too effective). Troops of the day also felt the SMLE was unfit for combat as well.
So, there's the beginning of research into new arms and a variety of calibre replacements for the .303, but there was also changes made to the .303 - the Spitzer bullet introduced in the Mk. VII variant changed the game entirely. Tactics and training were also seriously revamped - the 1909 Musketry Regulations being the most revolutionary change in doctrine the British Army ever underwent, which included the wholesale adoption of the SMLE as the primary weapon of all troops (rather than a hodgepodge of rifles, long ones for infantry, carbines for cavalry, etc).
The Pattern 1913 rifle, which could have replaced the SMLE early in her life ran into issues with the .276 Enfield cartridge with excessive flash and recoil, overheating and barrel wear. Obviously this didn't stop the Boffins from trying to find solutions to what ailed the .276 Enfield, but the Great War intervened and saw the SMLE kept on.
What I'd be interested in finding out, was whether or not the British ever dabbled with .276 Enfield or any other replacement calibre(s) during the interwar period?