The wording doesn't really cover this situation and I see how people could claim that less than a minute had passed between the spell being cast and the DM asking for the roll but that is crossing over between in-game and out-of-game contexts.
A common sense approach to the question says no, as the examples you provided in the question are long, potentially several hours in duration and neither spell has a duration even close to what you would need.
There isn't one moment in a forced march when a character becomes exhausted, nor is there one moment on a night watch where the character chooses to observe their surroundings.
For example, how would the PC with the Resistance cantrip know to cast it on you for an exhaustion saving throw? Even if you are the PC, how would you handle that? If you wait for the DM to say "Roll a CON save" and you interject with "I cast Resistance!", that's some powerful meta-gaming going on.