Does it affect screen readers?

Screen readers ignore the view transition pseudo-elements and read the page behind it.

Navigation API:

Any navigation that is intercepted and converted into a single-page navigation using navigateEvent.intercept() will be communicated to accessibility technology in the same way as a cross-document one. Using intercept() serves as a opt-in to this new behavior, and the provided promise allows the browser to know how long the navigation takes, and whether or not it succeeds.

— Navigation API explainer

Though, in my testing this wasn’t always the case. I hope that’s just because this stuff is in the early stages. When implementing, do your own testing.