Hello. Times are changing. Employers are now more open to looking at what an individual has done in their professional and personal life rather than just looking at what University you have attended. In recent years, some employers have actually removed a university degree as being one of the criteria they use for considering applicants.
That being said, there is still a place for educating yourself in a formal environment and receiving the associated accreditation. Many employers are openminded to consider applicatants with no formal degree but the reality is many employers still make this a requirement.
Specific to your question Bham Uni vs Bham City, even though it is politically correct not to have bias, I believe there is still a huge amount of bias. Birmingham University has a much longer history as a university than Bham City University. Some employers may consider this a factor in assessing your application.
Even I’m a little biased too. I would say go to Bham Uni because that’s an institution I know. The reputation of the Bham ComSci department when I was applying was outstanding. I believe it was one of the 3 best departments in the country at the time. But City may have something special that suits your personal circumstance much better.
In terms of making your choice, well I would research the league tables on Google. This is exactly what the HR departments of the organisations you will be applying to will be doing.
If you can, I would also try and visit both campuses and talk to faculty staff. If this is not possible, reach out to as many people from each institution as you can (past and present) to get their view on what each university is like. You can even search for people on LinkedIn who have attended each Uni and ask them what it was like (just like you’re asking my opinion here).