For instance, with a normal passport, the United States requires it be valid for a specific amount of time before traveling to the desired country. In this case, I'm trying to travel to Canada, but I'd like to avoid waiting three months before travel. Is Nexus similar in anyway, or once the application process is over am I free to travel right away?
I think you've got it backward: Some countries require visitors' passports to have X months validity remaining before their expiration dates. Not that the passport must be valid for X months from the time it was issued. But neither the US nor Canada require this of each other's visitors.
In any event the NEXUS card is valid as soon as you receive it, which owing to the unidirectional nature of time will be after its date of issue, and you can use it the same day.