I see this happening every time I switch on the lights in my bedroom. They are 4 downlights; 2 of them are old halogens and the other 2 have been replaced by LEDs. The LEDs switch on (and especially off) much faster than halogens.
The reason is that the halogens have a fat filament that takes the best part of a second to get to full brightness; switching off takes several seconds. The LEDs on the other hand are semiconductors that switch on or off in microseconds. Overall, the time difference is easily noticed.
Of course, my bulbs are in parallel but putting them in series would give the same effect. If the bulbs in series are the same type, then they will light up almost simultaneously. The only difference is the time taken by the electric field to travel down the wire. At close to ligthspeed, that is far too fast to be noticeable.