The standard date for the adoption of the Roman alphabet for English would be 597 when Augustine arrived in England to become Archbishop. Under his direction King Athelberht of Kent issued a law code written in English using the Roman alphabet in the first few years of the 7th century. The earliest dated English runic inscription is on the Undley Bracteate from the 5th century -- much earlier. This inscription is distinctively English because it contains the ᚩ (os) rune for the phoneme / o /, a distinctly English runic form adapted from ᚫ (ansuz) because of standard sound changes in the languages.

To go briefly into some phonology, the phoneme / a / in Proto-Germanic (sounds like the "a" in father) developed into the phoneme / æ / (like the "a" in cat) in Old English. This meant that the rune ᚫ (ansuz) now had the / æ / sound. The English renamed it from ansuz to æsc (pronounced ash). But now they had two further problems. First, the Germanic name for the letter ᚫ was ansuz (a word that means god). The / a / in this word was different than others because it was followed by an / n /; these / a + n / phonemes did not become / æ / but instead became / o / (and then the / n / disappeared). So in Old English, ansuz was now pronounced os (endings like -uz also disappeared in Old English). So now the early English needed a new rune for the phoneme / o /, so they took the ᚫ and put two upticks on it, forming ᚩ . The second problem is that when they used ᚫ for the / æ / sound, they no longer had a letter for the / a / sound, so they altered ᚫ again, creating ᚪ, which they used for / a /.

Similar alterations caused by sound changes happened throughout the runic alphabet as used by the early English. The Norse languages underwent their own sound changes which also caused some alteration of the earlier Germanic runes to fit their phonology.

You can find a lot of information on runes in Wikipedia, but also in books like Ray Page’s An Introduction to English Runes.