From BSA Wiki:
The BSA factory at Small Heath was bombed by the Luftwaffe on 26 August 1940 resulting in one H.E.bomb and a shower of incendiaries hitting the main barrel mill which was the only one operating on service rifles in the country, causing the unaffordable loss of 750 machine tools but fortunately no loss of life. Two further air raids took place on 19 and 22 November 1940. The air raid of 19 November did the most damage, causing loss of production and trapping hundreds of workers. Workers involved in the works Civil Defence were brought in to help search for and clear bodies to get the plant back into production. The net effect of the November raids was to destroy machine shops in the four-storey 1915 building, the original 1863 gunsmiths' building and nearby buildings, 1,600 machine tools, kill 53 employees, injure 89, 30 of them seriously and halt rifle production for three months.
The Government Ministry of Supply and BSA immediately began a process of production dispersal throughout Britain, through the shadow factory scheme. Factories were set up at Tipton, Dudley, Smethwick, Blackheath, Lye, Kidderminster, Stourport, Tyseley, and Bromsgrove to manufacture Browning machine guns, Stoke, Corsham, and Newcastle-under-Lyme produced the Hispano cannon, Leicester and Studley Road produced the Besa machine gun, Ruislip produced the Oerlikon 20mm cannon, Stafford produced rocket projectiles, Tamworth produced two-pounder gun carriages, Mansfield produced the Boys Anti-tank gun and Shirley produced rifles.
These were dispersal factories which were in addition to Small Heath and the other BSA factories opened in the two years following the 1940 blitz. At its peak Small Heath was running 67 factories engaged in war production. BSA operations were also dispersed to other companies under licence.
Note - No further mention is made of Dudley ever being used!