To summarize it in a few words: It was uncomplicated, but lenghthy (which made me tired).

To begin with, I only slept for four hours, and my adrenaline levels were pretty muchlatively high, although I am used to this.

Then, I arrived at my school, waited for a bit, entered the classroom, had to show my passport (Cambridge requires Photo ID), and took the first sectiontest of the day: Reading and Use of English.

It was okay, - although I was not sure on what I wrote at some parts, my teachers advicsed me not to leave any blank spaces.

It's a little ambiguous here if the contrast is intended between the first clause and second clause, or between the second clause and last clause

SThe second section, which was Wwriting, was relatively easy; I had to write a mandatory Eessay on road safety, and then I chose to do a review on a fictional Ssports television channel.

If the title of the section was "Writing" (imagine a heading at the top saying "Writing") you could maybe use the title case version here, but it looks a little weird because how it's used in the sentence makes it look more like describing the content. I didn't correct it in the first sentence as I wasn't sure if that was an exam title or not, but it feels here to be used as a topic.

"essay" definitely shouldn't be capitalized though, because in context you're just talking about the essay you wrote, even if the exam section was titled "Essay"

I nearly missed the third sectiontest, Listening, because I was distracted (by) having a chat with my English teacher, and I couldn't bring or use my smartphone at the time.

The "by" here will usually be omitted in speech, but is usually included in writing.

"I nearly miss" would be describing something you regularly do. "I nearly missed" is for something that (nearly) happened in the past.

I personally dislike listening tests, and I think this was the hardest part, because I had to listen to British accents despite us studying American English for our entire lives.

It was in groups of two people, and the other person in my case iwas a friend of mine, so no worries.

The thing is, for some reason, a thirty-minute delay happensed: The couplepair that had to take this part at 14:00 (2 PM) instead did it at 14:35 (2:35 PM).

Couple has the implication of a romantic couple, pair is more neutral here.

Our turn was at 15:25 (3:25 PM), nonthelesshowever we were summoned at 15:55 (3:55 PM).

The word you were trying for here is nonetheless, however it sounds a bit unnatural here.

It took less than 20 minutes and it was normal, although this skill is what I lack the most since nobody at homes can speak English.

Normal doesn't really say anything here. Depending on what you were aiming for, one of the following might be better:

"as expected" - this means the test went how you expected it to go
"of medium difficulty" - if you're just pointing out it's in the middle of the range of difficulties it could have been

Well that's it, I'm tired.

To summarize it in a few words: It was uncomplicated, but lenghthy (which made me tired).

To begin with, I only slept for four hours, my adrenaline levels were pretty much high, although I am used to this.

It was okay, although I was not sure on what I wrote at some parts, my teachers advicsed me not to leave any blank spaces.

advice = noun
advise = verb

SThe second section, which was Writing, was relatively easy; I had to write a mandatory Essay on road safety, and then I chose to do a review on a fictional Sports television channel.

I nearly missed the third section, Listening, because I was distracted having a chat with my English teacher, and I couldn't bring or use my smartphone at the time.

I personally dislike listening tests, and I think this was the hardest part, because I had to listen to British accents despite us studying American English for our entire lives.

The thing is, for some reason, there was a thirty-minute delay happens: The couple that had to take this part at 14:00 (2 PM) did it at 14:35 (2:35 PM).

When speaking, it's common to say "Thing is..." but it's too casual for most writing.

Our turn was atscheduled for 15:25 (3:25 PM), nonetheless we were summoned at 15:55 (3:55 PM).

"at" is fine grammatically, but "scheduled for" makes the meaning more clear. When I read the first part of your sentence, I thought you actually got to go at 15:25.

It took less than 20 minutes and it was normal, although this skill is what I lack the most since nobody at homes can speak English.

Well that's it, I'm tired.