I just got a 8bitdo ultimate bluetooth controller which I'm trying on Cyberpunk 2077. Unfortunately A is switched with B, Y is switched with X, as the layout is set for something like an Xbox controller and 8bitdo is different.

Can it be changed?

Yes. Download the 8bitdo Ultimate Software (https://app.8bitdo.com/). Plug in your controller via USB. Boot up the software and click "Windows". On the right hand side you'll see the "button" and "command" headers. The "button" side is the physical button on your controller, and the "command" side is what the computer will register the input as. Click the drop down menu on the command and choose X for X, A for A, Y for Y, and B for B. After that click "sync to device" and you should be good to go. You are going to see a small pill shaped button in the middle of your controller. If your inputs are still being registered incorrectly, press that button and try again.

Edit 1: typo.

Edit 2: correction

Thanks very much. I am trying this out.

It works well except I can't seem to run with the controller. I'm having to hit shift on the keyboard to get running.

I'm not sure what this could be. You should probably contact support.