This is a pretty good story, but there are places where it can be improved. Firstly, concerning your dialog, you seem to be doing something that I did until a teacher finally corrected me. When you're writing someone's dialog, you treat the quote itself as a separate sentence from the body of the story, rather than as a part of the narrative. For example, you wrote
“No time.” He muttered.
The problem with that, is that it's two different sentences. One being 'no time' and the other being 'he muttered.' Basically, 'no time' is thrown out there, and then he muttered - muttered what? It should be
"No time," he muttered.
That way, it's clear that 'no time' is what he's muttering. It didn't make much sense to me at first, but when you pause after the periods when reading you should see what I'm saying.
More specifically to this story, there were a few things; first, your dialog seems a little unnatural in places. To me it seemed that it was written more in the vein of the narrative than that of someone speaking (and right at the beginning you have 'who is to' which struck me as incredibly out of place given the way the guy speaks otherwise). The conversation between husband and wife seemed a little off to me, too, so I think it could use a going-over, maybe be fleshed out a little. Particularly in the fact that she is stated to have read that there's a strange disease affecting millions while her husband is talking about their mortality...seems like she might be able to put two-and-two together.
I'm fairly certain that it was part of your plan to have no real demarcations between time frames and whatnot, but it's a bit jarring while reading, and I honestly only got it because you mentioned something about it in the summary.
The line at the end, in bold, seems out of place and extraneous to me, and I think the line before it is a much better ending, but I guess it's supposed to go along with the title rather than as part of the story itself?
And, uh...why does his watch have no hands?

That was sort of weird, and sad. It was written nicely.