Personally I'd use SECS_PER_MIN, SECS_PER_HOUR, etc. I've even been known to use NANOS_PER_SEC on occasion. I would always do that if a language lacked scientific notation for integer literals.
It's not about readability, exactly. The reason for using SECS_PER_DAY rather than 86400 is not just to do with what general knowledge we expect of the reader. It's about self-documenting code, which means unambiguously showing your working.
Sure, everyone knows there are 60 seconds in a minute. But they also know there are 60 minutes in an hour. If you use a literal 60, it's probably obvious what your code is intended to do. But why take the chance?
If you use SECS_PER_MIN, then it is definitely obvious that you're converting a time in minutes (just 1 minute, in your example) to a time in seconds. You are not, for instance, adding one hour to a time measured in minutes, or one degree to an angle in minutes. You are not adding 5 feet to a length measured in inches.
Named constants provide more context for the surrounding code. For your example of adding a time, we know just by looking at one line that $x needs to be a time in seconds. The name of the constant reiterates that $x is not a time in millis, or clock ticks, or microfortnights. This makes it easier for everyone to check and maintain correctness of units: they can tell by looking that what you intended to do is what you actually did. They never have to even entertain the notion that you intended to add 60 millis to a time measured in millis, but got it wrong because $x was actually in seconds.
Named constants also help avoid typos. Sure, everyone knows there are 86400 seconds in a day. It doesn't necessarily follow that they won't typo this as 84600, or that they will immediately spot the error if someone else has. Granted you have complete testing, so such an error would never make it into the field, but the most efficient way to fix it is to prevent the faulty code making it to test in the first place. So I'd define the constants like this (or whatever syntax):
SECS_PER_MIN := 60;
SECS_PER_HOUR := 60 * SECS_PER_MIN;
SECS_PER_DAY := 24 * SECS_PER_HOUR;
SECS_PER_WEEK := 7 * SECS_PER_DAY;
Or, if the other constants were needed anyway (which in the case of time they probably wouldn't because you normalise everything into secs at the first opportunity to reduce the chance of confusion):
SECS_PER_MIN := 60;
MINS_PER_HOUR := 60;
HOURS_PER_DAY := 24;
DAYS_PER_WEEK := 7;
SECS_PER_HOUR := SECS_PER_MIN * MINS_PER_HOUR;
etc.
Note the order on the RHS: the minutes visibly "cancel", making the working even more clear. Not such a big deal with time, but it's good to establish a consistent scheme early, so that when things get nasty later (CLOCKS_PER_SEC, PLANCK_LENGTHS_PER_PARSEC) you can get it right using familiar techniques.