The origin of month is Old English monað from Proto-Germanic *mænoth-, which is related to *mænon- ("moon"). A month as a unit of time based on the cycles of the moon was invented by the Mesopotamians, but there is evidence that humans have been counting days relative to the moon since the Paleolithic age.
Before adopting the Julian calendar, English speakers in Britain followed this 12-month calendar:
1st month: Æfterra Geola - "After Yule"
2nd month: Solomonað - "Sun Month"
3rd month: Hreþmonað - from the divinity Hrepe
4th month: Eastermonað - from the divinity Eostre (probably where the word Easter came from)
5th month: Ðrimilcemonað - "Cow Milking Month"
6th month: Ærra Liþa - "Before Liþa"
7th month: Æfterra Liþa - "After Liþa"
8th month: Weodmonað - "Weed Month"
9th month: Haligmonað - "Holy Month"
10th month: Winterfylleð - "Winter Month" (first full moon of winter)
11th month: Blotmonað - "Sacrifice Month"
12th month: Ærra Geola - "Before Yule"
It is unclear when exactly English speakers adopted the Julian calendar, but it was certainly no earlier than 45BCE, because that's when it was invented. There are clues in the etymologies of the modern English months because they are based on the names of the Julian calendar months: Ianuarius, Februarius, Martius, Aprilis, Maius, Iunius, Quintilis (later Iulius), Sextilis (later Augustus), September, October, November, December, Intercalaris (used only in leap years).
It appears the we adopted September first, and an exact first attestation is not pinned down. Many of the other month names were adopted between 1000AD and 1100AD, and the rest by the late 14th century.
In September 1752 Great Britain and its dominions adopted the Gregorian calendar, which is the internationally recognized civic calendar toady.
For more calendar-related info, see Calendopedia.com