Workers were not given a choice between the military and civilian work.
Men between 18-41 were liable for military service. When you were called up, you went. The only exemptions were conscientious objectors and people employed in reserved occupations. Reserved occupations would certainly include armaments factories, though whether it included the ones your relatives worked at might need further research by you to confirm this.
Recruitment for all factory work was a huge problem. Young men called in to the army were no longer available to be recruited. Non-essential factory workers were called up, increasing demand for trained people across industry. Consequently many women entered in to this type of work who might never have worked at all. Much research has been done on the actual effect that this lack of available workforce had. One effect was to push up wages. This extra money might have been what attracted your family to Birmingham.
However, rural areas, including farms, were also affected by lack of labour, which is why women were recruited in to the Womens' Land Army. The WLA was voluntary from 1939, but conscripted from 1941. I think that factories generally paid better than farm work. So, work as such would not have been the point, but money would.
Of course, we can't know if other things affected their decision, maybe moving nearer relatives, or redundancy due to war dis-location. Many businesses moved north to avoid enemy attack.
Do you know how old these relatives were, or whether they were men, women, or both, who worked in these factories ?
Do you know what they did for a living before they moved ?