I don't see how they would given that they were working on the .276 Enfield. At 'best' they could adopt the 7mm Mauser, the inspiration for the .276 Enfield. 6.5mm Arisaka was against all conventional wisdom of the time due to combat experience to that point. Say they do that early enough, say 1910 as you suggestion, but honestly earlier would be better and since 7mm Mauser had been around since the late 19th century it could have been adopted several years earlier. There is no reason it couldn't have had a ~140 grain bullet like the Spanish 7mm Mauser spitzer adopted in 1913, which would have make it very flat firing, longer range, and considerably milder recoiling than 174 grain (despite having an aluminum or paper filled tip) flat base .303 bullet of WW1.
Though not as mild recoiling as the Arisaka round and with a hotter loading, it doesn't need as long of a barrel to reach peak performance and would have been probably quite controllable in the 1918 BAR, though even with a heavier barrel heat build up will be an issue, just not as much of one as the .30-06 model. Even better the base diameter is the same as the .30-06, so conversion is very easy.
Per the below table 7mm Mauser with 140 grain bullet has only 2/3rds the free recoil energy of the .30-06 Springfield.
https://www.chuckhawks.com/recoil_table.htm
That said Browning wouldn't make it in 7mm Mauser, it was only for US force, they didn't share with the Brits.
Now the Lewis Gun in 7mm Mauser is very doable and with a lighter powder load it won't heat up as quickly. Plus if Lewis and/or the Brits get their act together early and think up the designation they could have made the Lewis Assault Phase Rifle in 7mm Mauser:
http://www.smallarmsreview.com/display.article.cfm?idarticles=2192
A weapon lighter and lower recoiling than the OTL BAR even though it used the same caliber:
Doesn't seem that much different from the Lewis Gun without the barrel shroud, just with the magazine below rather than on top:
https://www.rockislandauction.com/detail/64/1591/savage-arms-corporation-lewis-machine-gun-3006
View attachment 470191
Seems like if you change the meaning of 'assault rifle' the Lewis assault phase rifle would be the WW1 assault rifle.