The resistance of a filament increases with temperature. So a bulb that is already lit up has a larger resistance than a cold bulb.
In this video, when the 10 bulbs are all lit up from cold together, because they grow hot together at the same rate, their resistances are the same at any one time. So the potential difference across each bulb is always 1.2 V (12 V /10 bulbs).
When 1 bulb is lit up from cold, with all the other 9 already hot, the initial potential difference across the cold bulb is much less than 1.2 V. This is because the other 9 hot filaments, with their larger resistances, has more than 1.2 V across each of them, as the potential divider principle dictates.
The pd across the 10th bulb will however eventually grow to become 1.2 V, when the bulb has heated up and its resistance becomes the same as all the other bulbs.
Compared to a bulb with 1.2 V across right from the beginning, a bulb with pd increasing slowly towards 1.2 V takes a longer time to light up.