Loss of speed from age doesn't impact an image that's already been captured on the film. Fog, however, does. The main reason to process the film as soon as possible is because it's no longer in its factory-sealed container, may have been exposed to moisture or other conditions that are bad for the film. The film can only possibly degrade from that point on (as in, it can't get better) so there's no point waiting to develop it.
This roll of film was in a bunch of stuff I bought. It was Tri-X 35mm, shot who-knows-when. I developed it - but the problem was it was likely pushed and the negatives ended up very thin. If I'd developed it a few minutes longer, it would have been much better.
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I also developed a film pack that was shot in the mid 50s. The images were on there fine but the emulsion suffered from exposure to the paper and age fog.