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We know that battery produces charge separation between the two terminals by means of some chemical reaction. So now we have negative charge (accumulation of electrons) at negative terminal and positive ions at the positive terminal, overall the battery is neutral.

Now the question arises how the elections flow when we connect this battery in a circuit, is it the reputation of the elections in the wire from the negative terminal and these repelled electrons get accepted by positive ions at the positive terminal?

If so, this concludes that if we connect a piece of wire only to the negative terminal we would get a current for very short period of time because the electrons get repelled in the wire and at the end of the wire we will get the same amount of charge collected as at the negative terminal i.e we get the same voltage when we probe at the end of wire to positive terminal as the battery's itself.

Now is it also true for positive terminal or positive ions at positive terminal will only accept the electrons when the circuit is complete because overall the battery is neutral and positive ions in the wire can't be repelled through positive terminal, only electrons can be accepted at the positive terminal.

If the above argument is correct, then where comes the drift velocity and isn't it looks like diffusion of electrons as near negative terminal wire have greater concentration of electrons?

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If you connect a single wire to a battery, there will indeed be some charge redistribution (a short-lived current) to the wire. The single wire acts like a capacitor. However, since the voltage is usually low for a "battery", and the gap between the wire and the other terminal of the battery is comparatively large, this charge is very small. But of course, it is sufficient to be able to detect the voltage of the battery at the end of the wire.

Like you already have recognized correctly, it does not really matter if it is the positive or the negative terminal, if there are at least some mobile charge carriers for the current.

Because the redistributed charge is very small, especially compared to the total concentration of electrons (i.e. even those that still get neutralized by positive atom bodies), the difference in electron concentration is negligible and hence, there is nothing worth mentioning with respect to drift velocity.

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