Yeah -- nothing mysterious is happening, the electrons coming out of the battery are just causing an electric field, which causes current in the wire near the 'lightbulb.' This electric field obeys the c speed limit, so if the battery terminals were far away from the 'lightbulb,' there'd be a fundamental physics limitation, the current couldn't start until at least d/c (where d is the distance).
More likely the electric field would be too weak to measure, but that's not fundamental -- you would just have trouble finding a sensitive enough 'lightbulb.'
