So games in which there cannot be a tie have no Nash equilibrium?

No, that's not a general rule. It's just the case that in this particular game, if you're losing you always have a better option that can be achieved just by changing your own strategy. If your prospects for improvement relied on others changing their strategies too, then you could lose and still be in a Nash equilibrium. (For an example of such a game, see battle of the sexes)

I thought the requirement was that there's no way to do better with an alternative strategy.

Sort of. It's that there's no way to do better with an alternative strategy, given perfect knowledge of others' strategies.

I was also assuming that everyone guesses at the same time

They do in the actual game; it's just that that's not relevant to evaluating what counts as a Nash equilibrium.

But this means it's no longer a perfect-information game, and that there's not going to be a Nash equilibrium.

I'm not entirely clear what you mean by the first half of this sentence, but the conclusion is false. Even if everyone guessed in turn, there would still be a Nash equilibrium with everyone playing zero.

Thanks for your patience :)

No problem. ;)