I doubt it. By the time Rome finally collapsed it had been in decline for a couple centuries. Populations declined throughout the Italian peninsula during this period, that's why it was conquerable in the first place. By the time the Ostrogoths overran Italy the peninsula was a shadow of its former self and there were few people left to flee. That, and the Ostrogothic Kingdom functioned nominally as a suzerainty of the eastern empire so there wasn't a tremendous change in the daily life of the Italian peoples.