MC is 10 years old when he is first tested for magic. There is a bit of adventure with the noble's rescue and the alchemist's village and then they travel to the old tower and from there the capital. And now he is 12. Did they really travel for 2 years? I thought it might have been a typo at first but he really is 12!
Yeah I think it was 5. He got to the tower and then sorted everything out and then they timeskiped his training to be a mage, so the timeskip was just the years he was required to complete to become an official mage. Though it was been a while since I read it so I could be wrong.
He was declared as a mage during his first test itself and that was when he was 10 and then he was declared the youngest archmage at the age of 12. The chapter in between does not seem disjointed and do not suggest a timeskip. Edit: chapters in between do not*