Use of "minute or two" to mean "moment" in the early 20th century?

It seems plausible, if not exactly likely, that Tommy might want to wait for a full 60-120 seconds; in context, he's doing something sneaky, and because he presumably doesn't want to get caught, he would be justified in being cautious.

That aside, your assumption that "minute" isn't a definite amount of time seems fair. And the use of "minute" would be colloquial, not amateurish or mistaken.

This ngram shows that the use of "moment" was declining while Christie was writing, and has declined markedly since her death. If anything, her substitution of "minute" for "moment" seems to anticipate a broader trend in the language.

According to the OED, as early as 1393, a "minute" was

The space of a minute (also minute while); a short space of time, an instant, a moment.

If usage of minute for moment was acceptable then, I think it's safe to assume it was fairly standard in Christie's time.