What are the accessibility implications of requiring a button press or time-out on a game over screen?

The accessibility issue is really only a problem when the input for leaving the game over screen is different form the input required to play the game.

For example, when the player is unable to operate more than 3 buttons comfortably, and your actual game only uses 3 buttons, then they can play it just fine. But when you then ask for a 4th button to leave the game over screen (or worse, a different input device like a mouse) then you added an unnecessary accessibility hurdle.

But as long as the input action which leaves the game over screen is the same as an input action which is critical for playing the game, then you are fine. Now any player who is able to play is able to leave the game over screen. And besides, it's more intuitive and convenient for every player to reuse a known input method than to introduce a new one. Another example for how thinking about accessibility improves the UX of everyone.

On the other hand, a game over screen which does have an automatic timeout can become an accessibility problem in itself. When you add some more information to the game over screen, then the player might need some time to absorb that information. The player might also want a break before starting the next round of the game, so pausing on the game over screen is a natural point for that.

It depends as well on your type of the game. For some games the game over screen gives you an overview of what you achieved, your progress, maybe even an option of retry current level, going back to level selection or main menu. This still holds true for simple games.

Another factor is that your game over screen can function as a break to reset your mentality, prepare for another round or reflect on your last game.

Finally as a user, I feel annoyed when I'm forced to wait. If you have a timed game over screen, allow the user to skip/ shorten it with the same user control they are used for playing your game.

As @BernhardBarker stated

If you have difficulty pushing buttons, you may have difficulty playing most games.

I would go with the approach that matches your whole game controls. If you control your game using buttons, wait for button press. If you control game using voice commands, use voice commands for continuing. If your game has high contrast and big elements for vision-impaired players, use them on game over screen. And so on and so forth.