During the last years of the Roman Empire, there were invasions from many "barbarian" tribes, such as the Franks, Lombards and Visigoths. In modern day France, Italy and Spain Latin-based languages eventually took hold, but at what point did these languages develop?

I'm particularly interested because of the contrast of these countries to England, which was also invaded by non-Christian Germanic tribes. In England, the language remained Germanic, whereas on the continent the languages of the invaders changed drastically. This is presumably down to the influence of the Catholic Church, but I find this intriguing because the clergy in these countries would have spoken Latin, but probably not the invaders themselves.