English was written in the Latin alphabet even in the Old English period. Latin letters were introduced in the OE period by Irish missionaries around the 7th-8th century. The Anglo-Saxons converted to Christianity relatively early, and literacy was a consequence of conversion, so the populace already was predisposed to the Latin alphabet given that that is what is used in the Bible. There is no record of a "struggle" to overthrow futhark.
There is variation in ancient manuscripts, some of which is plain copying error, some being dialect differences including change in the language. Some variation reflects orthographic practices, such as the introduction of letters not originally in Latin (some taken from futhark: ð,æ,þ,ƿ (wynn), as well as macrons to mark stress, j, w when they were invented, velar vs. palatal dots (ċ,ġ) etc.
Literacy and orthography was top-down, primarily driven by the church, much later under the post-Norman regime secondarily driven by the bureaucratic functions of government. English does appear in some manuscripts. Generally, though, Latin and then Anglo-Norman / French were the "official" languages, with English officially re-emerging in 1258.