How long did Roman culture and the use of Vulgar Latin (or some type of North African Romance language) survive in North Africa?

First off, it's worth noting that this will depend on the specific region of North Africa. So, for example, there's very little evidence of Christianity, Latin, or any kind of Roman culture in the Libyan interior. There are also relatively few Latin inscriptions from western Algeria.

I'm going to focus on an area roughly corresponding to modern Tunisia and eastern Algeria, where the Islamic conquest seems to have absorbed a sizable Latin-speaking population. As well as the usual funerary inscriptions and so on, there's some very interesting evidence for that in the form of early Arab coins minted in Kairouan. These were inscribed in Latin, somewhat clumsily translated from Arabic (they omit short vowels in the Arabic style), and were likely intended as religious and political propaganda aimed at a Latin-speaking Christian population, since they bear messages like "There is no God but the one God and he has no associate like him."

That initial Islamic conquests in the 7th and 8th centuries did not immediately cause a major cultural shift in most of North Africa. Settlement was limited and Arab culture seems to have been confined to the major cities and other important areas until the migration of the Banu Hilal tribe in the 11th century, which resulted in the Arabization of a wider region. Most rural areas would have remained culturally and linguistically Berber until the Banu Hilal arrived. But I think it would be very debatable to what degree that meant "Roman" culture survived outside the cities, since the countryside would have been less Romanized in any case.

So what evidence is there for the Latin language surviving? In the 12th century al-Idrisi wrote that the people of Gafsa still spoke an Africanized form of Latin ("al-latini al-ifriqi"). There are a small number of 11th century Latin tomb inscriptions from Kairouan and Ain Zara. There are also some interesting accounts of the Fatimid ruler al-Mansur, who had a great interest in Roman ruins. In 948, he visited the Djedar tombs near Tiaret and had a Latin inscription translated. He similarly had a Latin tombstone at Sétif translated. In both cases he was able to locate a nearby Latin-speaker, which suggests that it was still spoken -- but they both provided noticeably poor translations, suggesting that it wasn't spoken well, or only spoken in the form of a local dialect (which gels with al-Idrisi's account).

So there's some evidence for the survival of a heavily localized version of Latin until the 11th or 12th centuries, when there was a cultural shift associated with the arrival of Arab groups like the Banu Hilal. This would have been accelerated in the 12th century, when the intolerant Almohad takeover caused many (most?) remaining Christians and Jews to leave North Africa.