Lets assume you have a voltage source. Let it be for ex. 10VDC and an ideal one. It has no other circuit connected than its output wires. The voltage source keeps a constant voltage between its output wires. The electric field lives wildly when you move the wires. If the wires are say 15cm apart each other, the average E-field strength between the wires is 10V/15cm ie. about 0,67V/m. That's because voltage sources keep the voltage, the field adapts to the geometry of affected materials.
Idealizing a source of electricity by thinking it as voltage source is often a good approximation because we get electricity from processes which give to free electrons certain energy. If the process happens to arm free electrons with energy = 10 eV it shows out as voltage = 10V. In batteries that energy is released by chemical processes where the electron structure of the materials change or as we say "a chemical reaction occurs"
If you connect your voltage source to a circuit the field pulls free electrons until a new balance is found and the electric field again gets its final form. Hopefully you know "the influence" ie. how electrons move in a conductor until the generated unbalance of the charge exactly compensates the external field. That doesn't happen immediately because changes in electric field propagate in the space only about 30 centimeters/nanosecond. That's the speed of light. During the transient an electromagnetic wave propagates through the circuit. It's in the space between wires, very little of it is in the metal.
There can be also slower settling processes than the electromagnetic wave which reflects forth and back and attenuates gradually due losses in resistances and radiation out of the circuit. Capacitors for ex. which get discharged through big resistors can have very slowly increasing voltages.
The speed of the electrons in the metal is very slow. There's no need to keep high speed because there are so much free electrons in metals that all reasonable currents are possible with less than 1mm/s drifting. Colliding electrons do not cause the current to come out from a wire, it's the electric field. Generally electrons do not collide, they are so small and they push each other away if someone comes too close. You should know that materials are mainly empty space between atomic particles.
All phenomenas in circuits are finally caused by fields and their interaction with materials. Voltage and current are only handy ways to wipe the fields out of our minds and keep the false illusion of simplicity (=it's all in wires) alive. It works to some degree, as we know, but for ex. microwave circuits cannot be understood with Ohms and Kirchoffs laws.