Where is the electric field in an \$LC\$ circuit?
The E Field across the capacitor plates will oscillate with the H field around the inductor wire as the right hand rule for inductor current.
The E field will be present as soon as a voltage is applied or current applied to charge to a certain preset voltage, not simply after the circuit is closed. before t=0 it is a static E field condition.
It will oscillate forever with any non-zero initial condition and lossless electrodes , wire, dielectric and magnetic materials if any.
I=dQ/dt is true for both parts which charge flow in an oscillating loop.  It is sinusoidal and V vs I are always 180 degrees out of phase in this LC closed loop.