I have a 10 year UK ban on my South African passport but I have a Zimbabwean passport as well with different name and date of birth. What will happen if I use this?
Zimbabwe and South Africa are both visa-national countries, so your question resolves to: "what happens if I apply for entry clearance where a ban is still in effect on a different travel document?" You might be thinking that nobody has ever thought of this before and that there's a loophole available.
The previous use and ban on your South African passport was accompanied by the capture and storage of your biometrics. When you apply for entry clearance with your Zim passport, you will need to enrol your biometrics. When those are put into the system it will find a match with your previous removal. At that point you will be called in for an interview so they can get a voice recording and video of you, and then you'll get a refusal notice in the mail. A second ban will kick in, but I don't know (and I doubt anyone knows for certain) if multiple bans are served concurrently or consecutively. So the only thing that would happen is the loss of the GBP 83 application fee, another ban, and the disclosure of your Zim passport details to the other 'five eyes' members and Schengen. That's the best outcome you could hope for.
The only other case is if your previous ban was the result of clandestine entry to the UK. In this case if you used your Zim passport to somehow either get a visa or effect a clandestine entry, UKVI would most likely refer you to the Crown Prosecution Service under Section 2 of the Fraud Act 2006 because you would be benefiting by accessing the UK whilst representing yourself with a different identity. That would leave you with a criminal record and possibly cause the Zim or SA government to revoke your passport(s). It can be a quick way to go from two passports to zero passports.