For some reason I feel like Roger Rabbit, in that bar...
Anyway, this topic has been discussed in very long threads on this forum - I believe in General Technical Chat - and you might want to read those 50-80 pages threads to get a full spectrum of possible interpretations.
I'll give you mine here in its 'short' form.
Charges are associated with an electric field, charges in motion are associated with an additional magnetic field. Changes in the configuration (of position, speed, acceleration) of the charges propagate at the speed of light from the charge themselves, in the surrounding space.
When the switch is open, charges of opposite sign are accumulated at the switch's terminal - they are on the surfaces of the conductors the terminals is made of - and in part on the surface of the remaining conductors in the circuit (whatever is needed to have a net electric field equal to zero inside the conductor). When you flip the switch, the surface charge distribution start to change - some charge recombine, some surface charges move, some inner charges start to move. With this change in configuration, comes a change in the electric and magnetic fields that propagates at the speed of light in the medium.
You have two phenomena going on: the redistribution of surface charge along the conductor starting from the switch, and the perturbation in the field in the whole of space around the disturbed region. The first phenomenon will travel along the wires and will take a lot of time to complete when the wires are very long (even tho it is very fast in itself - just a fraction of the speed of light); the second phenomenon does not need wires and will travel through space and the insulating material at the speed of light (in the medium it is going through). This latter perturbation in the fields will start to change the distribution of charge in the conductors it encounter along its path, even if they have yet to be reached by the 'wired' perturbation.
This is the effect Veritasium sees at time 1/c, well before the perturbation on the wire reaches the lamp. It is an incomplete configuration of surface charge, that will be short lived unless the reinforcement will come from the perturbation of charge traveling along the wires.
Where is the energy traveling? This is debatable.
My view is that we put energy in the fields (or rather in the system of charges that produce them) by perturbing them, the perturbation travels at the speed of light and then we can extract energy from whatever portion of space has the right configuration of fields.
For example, by closing the switch we perturb the system in such a way that there will be an accumulation of charges of opposing sign at the resistor - conductor interfaces. These opposing charges will create a strong electric field inside the resistor, that will greatly accelerate the free electron, locally giving them a lot of energy that they will lose to the lattice, producing heat. The electrons did not have that energy before entering the high field region of the resistor, and they will not have it after leaving it. It's locally acquired, thanks to the EM fields that mantain that charge imbalance.
But for the EM fields to be that way, you need the conductors to host surface charge. That's why you need wires to 'transfer' large amount of energy.