What are the promises referred to by Peter in 2 Peter 1:4 through which we can partake of the divine nature and escape the corruption that is in the world by lust? What are the views regarding this according to Reformed Theologians and those of the Keswick theology?
The first part of this answer comes from Calvin's Commentaries on 2 Peter 1:1-4. Before we can address verse 4 we need to deal with the first three verses to put everything into context:
- Simon Peter, a servant and an apostle of Jesus Christ, to them that have obtained like precious faith with us through the righteousness of God and our Saviour Jesus Christ:
- Grace and peace be multiplied unto you through the knowledge of God, and of Jesus our Lord,
- According as his divine power hath given unto us all things that pertain unto life and godliness, through the knowledge of him that hath called us to glory and virtue:
- Whereby are given unto us exceeding great and precious promises: that by these ye might be partakers of the divine nature, having escaped the corruption that is in the world through lust.
Verse 1 - Like precious faith, through the righteousness of God:
This is a commendation of the grace which God had indiscriminately shewed to all his elect people... He adds, through the righteousness of God, in order that they might know that they did not obtain faith through their own efforts or strength, but through God's favor alone.
Verse 2 - Through the knowledge of god and of Jesus our Lord:
He [Peter] connects together at the same time the knowledge of God and of Christ; because God cnnot be rightly known except in Christ according to that saying.
Verse 3 - All things that pertain unto life and godliness:
So Peter does not speak hereof thenatural gifts of God, but only mentions those things which he confers perculiarly on his own elect above the common orderof nature... the common conditio of human life is not here referred to, but the peculiar endowments of thenew and Spiritual life, which derive their origin from the kingdom of Christ... he leaves to us not even the least particle of any virtue or merit. It was Peter's object expressly to ascribe the whole praise of our salvation to God, so that we may know that we owe every thing to him.
Verse 4 - Whereby are given unto us exceeding great and precious promises:
It is doubtful whether he refers only to glory and power, or to the preceding things also... the meaning will be, that first the promises of God ought to be most highly valued; and, secondly, that they are gratuitous, because they are offered to us as gifts. And then he shews the excellency of the promises, that they make us partakers of the divine nature, than which nothing can be conceived better... But we, disregarding empty speculations, ought to be satisfied with this one thing, that the image of God in holiness and righteousness is restored to us for this end, that we may at length be partakers of eternal life and glory as far as it will be necessary for our complete felicity.
Another Reformed Theology source of information concerning sanctification says this:
Two features are central to sanctification: Jesus Christ himself is our sanctification or holiness (1 Cor 1:30); and it is through union with Christ that sanctification is accomplished in us... . It is those who have received God’s promises who should purify themselves “from everything that contaminates body and spirit, perfecting holiness out of reverence for God” (2 Cor 7:1)  Calvin Institutes 3.1.1. Source: https://www.uniontheology.org/resources/life/the-reformed-view-of-sanctification
So, what are the "exceeding great and precious promises" that God gives "to them that have obtained like precoious faith"? All these promises have to do with our salvation. It is through those promises that believers are made "partakers of the divine nature. Here are a few:
When we believe on the Lord Jesus Christ, we are saved (Acts 16:31)
We experience a radical spiritual transformation which comes by virtue of becoming partakers of the divine nature (2 Peter 1:4)
We are made new creations in Christ (2 Corinthians 5:17)
We are born again, spiritually (John 3:3)
We die to our old sinful nature and our lives are hidden with Christ in God (Colossians 3:3)
As partakers of the divine nature, believers are made part of God's family (John 1:12)
We are conformed to the image of Christ Jesus (Romans 8:29; 2 Corinthians 6:18)
The greatest promise of all is that believers who have experienced the new birth have the power of the Holy Spirit indwelling them (Romans 8:37; John 14:16) and know that they will never be forsaken (Hebrews 13:5).
This is what empowers the born again believer to "escape the corruption that is in the world by lust".
24 Now I rejoice in my sufferings for your sake, and in my flesh I am filling up what is lacking in Christ's afflictions for the sake of his body, that is, the church, 25 of which I became a minister according to the stewardship from God that was given to me for you, to make the word of God fully known, 26 the mystery hidden for ages and generations but now revealed to his saints. 27 To them God chose to make known how great among the Gentiles are the riches of the glory of this mystery, which is Christ in you, the hope of glory.
The idea that Christ is being formed within the believer is the closest in my mind to the idea that we can share in the divine nature. God has communicable and incommunicable attributes, so there are limits.
Galatians 4:19: my little children, for whom I am again in the anguish of childbirth until Christ is formed in you!