Will diagonal PCB traces affect a sensitive CPU circuit?
There's no reason to use smooth traces at the frequencies you're working with. 45-degree corners are fine (they're done that way for manufacturing reasons, not SI necessarily.)
What is important, however, is that the Z80 signals have return paths. As you've shown, all your signals share only one ground, pin 29 of the IC. That's not going to work so well.
You can improve things by adding a ground plane to your board, and add some local bypass caps to VCC.