OP's First question
"So the main question is how does Jesus mediate"
For this reason he is the mediator of a new covenant,
It's important to remember
that this new covenant that Jesus is the mediator is for Israel and Judah only.
The old covenant is the compact made with Israel at Mount Sinai. It was two-sided where the people had to do their part and then Jehovah would do His. Since they failed miserably to ever obey Him, it made Him unable to fulfill His promise to bless them.
The new covenant which He will make with them after they have been restored to their land, and have received their Messiah, is radically different from the old. The people have no active part it whatever. All depends on Jehovah. Hence it will not be a failure. The law that was written on stones will be written on their hearts." Concordant Commentary
This is the covenant I will establish with the people of Israel after that time, declares the Lord. I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people.
This also goes along with Ezekiel 36: 22-29.
I will take you from the nations and gather you from all the countries and bring you into your own land. 25I will sprinkle clean water on you, and you shall be clean from all your uncleannesses, and from all your idols I will cleanse you. 26And I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh. 27And I will put my Spirit within you, and cause you to walk in my statutes and be careful to obey my rules.a 28You shall dwell in the land that I gave to your fathers, and you shall be my people, and I will be your God. Ezekiel 36:25-28
The new covenant with Israel as a nation has yet to go in effect. This will not happen until the deliver comes from Zion and Rescues Israel as it is written:
And so all Israel will be saved, as it has been written:
“The One Delivering will come out of Zion,
He will remove ungodliness from Jacob.
27And this is the covenant from Me to them,
when I shall take away their sins.” Romans 11:26-27
The way Jesus mediated His new covenant between Israel and God is made know in the Scriptures.
And Christ being come, chief priest of the coming good things, through the greater and more perfect tabernacle not made with hands — that is, not of this creation — 12neither through blood of goats and calves, but through his own blood, did enter in once into the holy places, age-during redemption having obtained; Hebrews 9:11-12
It is for this reason He is the mediator of the new covenant.
how much more shall the blood of the Christ (who through the age-during Spirit did offer himself unblemished to God) purify your conscience from dead works to serve the living God?
And because of this, of a new covenant he is mediator, that, death having come, for redemption of the transgressions under the first covenant, those called may receive the promise of the age-during inheritance, 16for where a covenant [is], the death of the covenant-victim to come in is necessary, Hebrews 9:14-16
On the other hand He is also
a mediator between God and all men. It is the man Christ Jesus.
God our Savior, who wills all men to be saved, and to come to the full knowledge of the truth; 5for [there is] one God, also one mediator of God and of men—the man Christ Jesus, 6who gave Himself [as] a ransom for all—the testimony in its own times— 1 Tim. 2:4-6
"Christ is Adams greater son,
a Man, And as such, is the mediator for the whole human race. Nor is Israel's God limited to them: For there is only one God For all mankind. The man Christ Jesus is giving himself as a ransom for all." concordant commentary
The way Jesus mediates between God and men is by giving himself as a ransom.
antílytron (from 473 /antí, "corresponding to, instead of/exchange" and 3083 /lýtron, "ransom-price") – properly, a full ransom, referring to Christ paying the complete purchase-price to secure our freedom (redemption) – i.e. Christ exchanging His eternal righteousness for our sin (cf. Ro 3:26; 2 Cor 5:21).
What is given in exchange for another as the price of his redemption.
So then, as through one offense to all men [it is] to condemnation, so also through one declaration of “Righteous” [it is] to all men to justification of life; 19for as through the disobedience of the one man, the many were constituted sinners: so also through the obedience of the One, will the many be constituted righteous.
OP's second question,
"the secondary question is whether the mediation is already done or ongoing?"
The Mediation in both cases has been secured and is still in process of being made known.
Through Jesus'-one act of obedience to the Father the course had been set for His complete mediation to be seen in the coming ages.
Definition of mediator,
one who intervenes between two, either in order to make or restore peace and friendship.
and through Him to reconcile all things to Himself—having made peace through the blood of His Cross—through Him, whether the things on the earth, whether the things in the heavens. Colossians 1:20