It depends.

If the game is already cast and VO/performance hasn't been entirely recorded yet, then yes, it will grind production to a halt.

If the game hasn't been cast yet, some studios might decide to go record VO in the UK instead. Typically that leads to lower quality unless the cast needs to be British. The American voice actor pool in the UK is much smaller than the U.S. for obvious reasons, and that leads to much lower quality of performance.

So basically, it will slow down productions if they don't have the option to go to the UK instead. For everything else, it depends how badly a studio wants to release within the given time frame, but there are options.