Learn English – What word to use here? To check the time/clock after short intervals of minutes /seconds

if you actually un-enrolled (officially withdrew from the school), you could say that you will re-enroll.

I took a break from university for 6 months to travel. I will re-enroll when I get back to my country.

If not (say you were on a leave of absence or sabbatical), you could say

I took a break from university for 6 months to travel. I will start back up again when I get back to my country.

or

I took a break from university for 6 months to travel. I will go back to school when I get back to my country.

Or, slightly more formally,

I took a break from university for 6 months to travel. I will resume my studies when I get back to my country.

This second suggestion is very similar to one of your ideas about how to express it. We don't usually talk about university as something one does, though. It is a place we attend, or go to, or it modifies some other noun. So your other suggestion could be modified to:

-"I am going to carry on with my university studies when I get back to my country."

-AmEng

Admittedly, I'm answering a BrE question as an American, but your source is suspect.

9.36 twenty-four minutes to ten

This is grammatical, but nobody in their right mind would actually say it. Who's got the time to calculate 60 minus 36 to come up with this version? You'd just say "Nine thirty-six". (If the time is close to a round value, it's perfectly normal to say it's "twenty to one" or "a quarter to three")

In the days of analog clocks, people would normally give the time to the nearest 5 minutes.

Now when the most likely way to find the time is to look at your phone, you'll mostly just read off exactly what it says, whether it's "ten fifteen" or "seven twenty-seven". When the minutes are less than 10, you'll add an "oh", as in "six oh five".