Several hadith state that Aisha was 6 when she married and 9 when the marriage was consumated. These hadith are narrated by Aisha herself and are considered strong hadith:
Bukhari vol. 7, #65:
"Narrated Aisha that the prophet wrote the marriage contract with her when she was six years old and he consummated his marriage when she was nine years old."
Sunan of Abu Dawud, volume 2, #2116"Aisha said, "The Apostle of Allah married me when I was six years old. He had intercourse with me when I was 9 years old."
There are others.
Also, the hadith state that Aisha was still playing with dolls when she moved in with Muhammed. According to this Islamic scholor, that would not be allowed if Aisha had her first menses:
"Al-Khattaabee said: From this Hadeeth it is understood that playing with dolls (al-banaat) is not like the amusement from other images (suwar) concerning which the threat (wa'eed) of punishment is mentioned. The only reason why permission in this was given to 'Aa'isha (may Allah be pleased with her) is because SHE HAD NOT, AT THAT TIME, REACHED THE AGE OF PUBERTY."
Source: The Beneficial Response Concerning the Islamic Ruling of Pictures/Images
by: Shaykh Abdul-Azeez Ibn Abdullah Ibn Baaz
A further reading of the hadiths narrated by Aisha show a child that was still swinging withher playmates and playing with her dolls moving, as a wife, into muhammed's house when she was 9 and having sex with Muhammed at that time.
Bukhari, volume 5, #234
Sahih Muslim, Book 008, Number 3311
Sahih Muslim, Book 031, Number 5981
Sunan Abu Dawud, Book 41, Number 4914
"The History of Tabari", volume 9, page 131
Encyclopedia of Islam, under "Aisha"
As to the question of climate, the main factors that influence start of menses seem to be home environment and nutrition. For example, the worse the nutrition, the older the onset of menses. Climate does not seem to be a factor. For girls in Bagdad the average age of first menses is 13.6. For girls in The Netherlands it is 13.3.
I have not done extensive research on this topic, but whenever I have asked a Muslim for evidence of this popular claim I get nothing at all.