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First of all, let me second Majenko in suggesting the use of an RTC. The Arduino Uno is clocked from a ceramic resonator, and these are nowhere accurate enough for proper timekeeping. C.f. the answers to How do I get an accurate time?.

Next, instead of counting seconds, I suggest you instead count milliseconds. I know it sounds like overkill, but it's actually simpler because you already have millis() returning a millisecond count:

void loop()
{
 uint32_t now = millis();
 static uint32_t last_minute;
 // Increment the time every minute.
 if (now - last_minute >= 60000) {
 last_minute += 60000;
 if (++minutes >= 60) {
 minutes = 0;
 if (++hours >= 24)
 hours = 0;
 }
 }
 // Do not advance the seconds when the user sets the time.
 if (time_is_being_set_by_the_user()) {
 last_minute = now;
 }
}

Oh, wait! No! Do not count milliseconds, instead count microseconds:

void loop()
{
 uint32_t now = micros();
 static uint32_t last_minute;
 // Increment the time every minute.
 if (now - last_minute >= 60000000) {
 last_minute += 60000000;
 // The rest does not change...
 }
}

Why? Because, as I said before, the ceramic resonator of your Arduino is not very accurate. You will eventually find out that it drifts significantly. However, if you compare to a good reference, you will be able to calibrate the drift. If, for example, you find that it drifts forward by 0.0025% (about 15 seconds per week, quite optimistic), you will be able to fix it simply by replacing 60000000 by 60001500. The same would work if counting milliseconds, except that you would only be able to do a coarse tuning. Oh, and do not worry about micros() overflowing.

First of all, let me second Majenko in suggesting the use of an RTC. The Arduino Uno is clocked from a ceramic resonator, and these are nowhere accurate enough for proper timekeeping. C.f. the answers to How do I get an accurate time?.

Next, instead of counting seconds, I suggest you instead count milliseconds. I know it sounds like overkill, but it's actually simpler because you already have millis() returning a millisecond count:

void loop()
{
 uint32_t now = millis();
 static uint32_t last_minute;
 // Increment the time every minute.
 if (now - last_minute >= 60000) {
 last_minute += 60000;
 if (++minutes >= 60) {
 minutes = 0;
 if (++hours >= 24)
 hours = 0;
 }
 }
 // Do not advance the seconds when the user sets the time.
 if (time_is_being_set_by_the_user()) {
 last_minute = now;
 }
}

Oh, wait! No! Do not count milliseconds, instead count microseconds:

void loop()
{
 uint32_t now = micros();
 static uint32_t last_minute;
 // Increment the time every minute.
 if (now - last_minute >= 60000000) {
 last_minute += 60000000;
 // The rest does not change...
 }
}

Why? Because, as I said before, the ceramic resonator of your Arduino is not very accurate. You will eventually find out that it drifts significantly. However, if you compare to a good reference, you will be able to calibrate the drift. If, for example, you find that it drifts forward by 0.0025% (about 15 seconds per week, quite optimistic), you will be able to fix it simply by replacing 60000000 by 60001500. The same would work if counting milliseconds, except that you would only be able to do a coarse tuning. Oh, and do not worry about micros() overflowing.