Writing won't be sufficient, given how Latin and Arabic diverged over time. This article about the dialects of Arabic is informative, and makes a comparison between early Latin diverging into French, Spanish, Portuguese, Italian, and Arabic now. Basically, Since the Islamic conquests Arabic has become mutually unintelligible between different regions which remained isolated from each other for long enough to evolve into unique dialects.
Often contemporary Arabic speakers will struggle or fail to understand each other without reverting to classical Arabic, just as Catholics in medieval Europe used Medieval Latin (again different from Eclesiastical Latin) for formal communication.
Perhaps the most interesting point made in the first article is that Arabic speakers will often revert to a common dialect, which is often Egyptian given the influence of Egyptian media in the Arab world. And this suggests that mass media, with radio, television, and the internet, has a potent effect. Not necessarily to unify dialects, but rather to make a common language more accessible.
Also worth noting the evolution of different accents between American and British English. Which is the first step towards a new language. Isolated accents become dialects, isolated dialects become languages. Of course that has come full circle with the domination of American English given American media influence, but it's still a decent case study.
People back in England noted the quirky new ways Americans were
speaking English within a generation of the colonists’ arrival. Over
time, the changes went beyond accent to include different words and
grammatical structures, adding up to a new dialect. Dialects have two
main causes. The first is isolation; early colonists had only sporadic
contact with the mother country. The second is exposure to other
languages, and the colonists came into contact with Native American
languages, mariners’ Indian English pidgin and other settlers, who
spoke Dutch, Swedish, French and Spanish. All of these languages
influenced American English, as did the English-speaking colonists’
origins in different parts of England, Wales and Scotland.
The speed at which this happens, and severity of language drift, will depend entirely on issues like immigration, prevalence of education, speed of transport, language institutions and restrictions (like the notoriously protectionist Academie Francaise) and how otherwise isolated communities are. Common written language isn't enough to prevent linguistic drift.
Keep in mind that Ottoman Turkey used the Arabic alphabet for centuries for Turkish, and Iran still uses the Arabic alphabet for Farsi. This complicates matters more, since neither language originated in Arabic, nor suited using the Alphabet. A common alphabet certainly isn't good enough in that case!