How do electrons travel between wires? I know that electrons can travel in metals when an electric field is applied and there is a potential difference created. This is due to the delocalized electrons which are free to move and have a small drift-velocity, and all of this is fine inside the metallic body; however, when we make a circuit, we join the ends of two wires through which electrons can move.
Now these ends of the wires are not metallically bonded, they are just 'physically' touching each other, so how do delocalized electrons from one wire transfer into another wire? I mean it's not like the delocalized electrons have a path to go from one wire to another, right? Do they 'jump' such small distances or do they follow some other mechanism? Do wires touching each other act like they are metallically bonded?
I have tried conducting some preliminary research on this question; however, I could not find much regarding my question.
 A: Let us take a simple example , a battery with two wires soldered so there is no surface between the pole and the wires. If we bring the two wires near each other , at a small distance a spark will happen ,a short, due to the migration of positive and negative charges over the distance between the two wires.
Contacts have been devised so as to be able to close the distance without sparking.The metallic surface of the contact,

allowing the charge transfer without sparking.
After this there are studies on what is called "contact resistance", from the abstract

A factor complicating the control of electric circuits is the contact resistance set up between two mating surfaces of metal which at times have to be separated in order to break continuity of supply. The two main methods of decreasing contact resistance-excluding the use of special contact tips-are to increase the contact area and to increase the contact pressure.

You ask:

Do they 'jump' such small distances

Well , it is quantum dimensions at the level of contact, one could call it jump,

Do wires touching each other act like they are metallically bonded?

Not really, as there is resistance to overcome.
From the abstract:

The two main methods of decreasing contact resistance-excluding the use of special contact tips-are to increase the contact area and to increase the contact pressure. Contact pressure can be increased only to a certain limit, beyond which it results in a welding-in of contacts. Large contact surfaces are uneconomical; they are, in fact, made as small as possible consistent with a contact pressure low enough to preclude welding-in.

If "welded in" then they are bonded.If the contact is movable  there will always be resistance
