The electric field travels through the wire at the speed of light, so there'll be like a billionth of a quadrillionth of a nanosecond delay before it hits the bulbs, unless your wiring travels all the way around the world.

If they are old-fashioned incandescent light bulbs, it takes some milliseconds for the filament to heat up enough to start glowing, the time is too short for human perception to notice but it does exist.

LED lights are all over the place, depends on how they are made, but even the cheap and nasty ones should have no more delay than a blink -- enough to notice, but not 6 or 7 seconds. I suppose that it's possible that somebody has made, and installed, special-purpose LED lights with a delay, but I can't imagine why anyone would do it.

What happens if you swap the bulb for a brand new one?