"Roll once per minute" vs "roll to see how long..."
 

In my games, I have replaced "roll once per minute" with a single roll that tells you how long it took or whether you failed. "Degree of success" (how well you succeeded) hasn't really been a thing in TFT AFAIK until we started talking about contests and opposed rolls. Here's how I do activities that take variable time:

1. Say how long you are willing to attempt the task -- you CAN'T abort early so be careful
2. Make your roll -- what you are rolling against might be secret
3. Your "degree of success" determines how long it took -- the degree may only be known to the GM
4. If the length of time is more than you were willing to spend in #1, you fail and, because of secrecy in #2, you might not know whether you even could have succeeded
5. In any case, failure almost always means you spent the maximum time for which you were willing to attempt the task, although a GM might rule that critical failure is immediate and spectacular

This plays well with other variable time rolls, like if you happen to use random encounters, the "time until the next random encounter", which you can also use instead of "roll once per hour" or whatever. Just one roll per encounter, the "degree of success" determines how soon the encounter happens. Watch out for critical success! If you are GM and you have already rolled for when the next random encounter will happen, try not to smile when a PC starts to pick a lock and sets the time limit too long...

Tasks might require several rolls of different natures, some of which could involve time like this.

Of note in rolls like this (and in contests) are the eight success classes:

Super critical success: 3
Critical success: 4
Automatic success: 5
Normal success
Normal failure
Automatic failure: 16
Critical failure: 17
Super critical failure: 18

Automatic success should allow the task to take the stated amount of time if the actual degree of success would exceed the time limit.

Automatic failure would make you reach your time limit, even if the degree of success is within the limit.

etc.