There were six of these 16/20 hp Milnes Daimlers, registered O 264-9, new in 1904. As you say, they were new to the Birmingham Motor Express company, one of the BET group companies later amalgamated to form the Midland 'Red'.
The Milnes Car Co. was the bodymaker, a tramcar manufacturer with factories at Birkenhead and Hadley, near Wellington, Shropshire (now part of Telford New Town) - the latter later became Sankeys foundry, making steel baths amogst other things!
They also built motor vehicles - or imported them (Daimler from Canstatt, Germany) and Milnes Daimler buses and lorries were fairly common among the few commercial motors around at the time.
They were not particularly reliable, and certainly not quiet - the final transmission, rather than chain drive as other makes used, was a gear-ended shaft which drove gear teeth on the inside face of the rear wheels. Road grit soon got into these exposed gears and wore them, there survive reports of constables ordering buses off the road for 'excessive noise'!
O 268 became a lorry in 1905, then a charabanc with registration O 1514.
O 264/7 became charabancs in 1907, registered O 2619/20.
All were withdrawn and sold later in 1907.
There were nine 24 hp Milnes Daimlers, too (O 1270-8) - slightly more reliable, but they went in 1907 as well. Here's one.