There are two different questions at work here, that you've kind of mashed together. The first question is "What is the speed at which a change in the electric field propagates?" The answer to that is the speed of light. In QED terms, the electromagnetic interaction that we see as the electric field is mediated by photons, so any change in an established field (say, due to shifting the position of the charge creating the field) won't be felt by a distant object until enough time has passed for a photon from the source to make it to the observation point.
The second question is "What is the speed of propagation of electric current?" This speed is slower than the speed of light, but still on about that order of magnitude-- the exact value depends a little on the arrangement of wires and so on, but you won't be far off if you assume that electrical signals propagate down a cable at the speed of light.
This relates to electric field in that the charge moving through a circuit to light a light bulb has to be driven by some electric field, so you can reasonably ask how that field is established, and how much time it takes. Qualitatively, the necessary field is established by excess charge on the surface of the wires, with the surface charge being generally positive near the positive terminal of a battery and generally negative near the negative terminal, and dropping off smoothly from one to the other so that the electric field is more or less piecewise constant (that is, the field is the same everywhere inside a wire, and the field is the same everywhere inside a resistor, but the two field values are not the same).
When the circuit is first connected, there is a rapid redistribution of the charge on the surface of the wires which establishes the surface charge gradients that drive the steady-state current that will eventually do whatever it is you want it to do. The time required to establish the gradients and settle in to the steady-state condition is very fast, most likely on the order of nanoseconds for a normal circuit.
There's a good discussion of the business of how, exactly, charges get moved around to drive a current in the textbook that we use for our introductory classes, Matter and Interactions, by Chabay and Sherwood. It doesn't go into enough detail to let you calculate the relevant times directly, but it lays out the basic science pretty well.
(It's a textbook for a first-year introductory physics class, so it sweeps a lot of condensed matter physics under the metaphorical rug-- there's no discussion of band structure or surface modes, or any of that. It's fairly solid conceptually, though, at least according to colleagues who know more about those fields than I do.)