As far back as the Exodus out of Egypt, God describes Israel as a kingdom of priests and a holy nation (Exodus 19:6). Later, Peter addresses Jewish believers in Christ: “But you are a chosen people, a royal priesthood, a holy nation, a people belonging to God, that you may declare the praises of him who called you out of darkness into his wonderful light” (1 Peter 2:9).
Peter then goes on to explain that this kingdom of priests, this holy nation, is not restricted to Jewish believers in Christ. Paul was commissioned to take the good news of the kingdom to Gentiles scattered around the Roman Empire. John wrote to the first century churches explaining that they – Christians both Jews and Gentiles – are included in this kingdom of priests:
To him who loves us and has freed us from our sins by his blood, and made us to be a kingdom and priests to serve his God and Father, to him be glory and power forever and ever! (Revelation 1:5-6)
In Revelation 5:9–10 it says God has redeemed people from every tribe and tongue and people and nation, and they are included in this kingdom of priests.
There are some Christian denominations who believe that God has finished with Israel and this kingdom of priests, this holy nation, is now “the Church”. Whether that embraces all Protestant churches, all Roman Catholic churches, and all Eastern Orthodox churches, I could not say. Just because a person self-identifies as a Christian does not mean they are automatically part of God’s kingdom of priests.
In order to see the Kingdom of God, a believer has to experience the new birth. This is a spiritual rebirth, an act of God whereby eternal life is imparted to the person who believes (2 Corinthians 5:17; Titus 3:5; 1 Peter 1:3; 1 John 2:29; 3:9; 4:7; 5:1-4, 18). John 1:12-13 indicates that being "born again" also carries the idea of "becoming children of God" through faith in the name of Jesus Christ. When a Christian is born again and spiritually renewed, they become a child of God by right of the new birth:
Therefore, if anyone is in Christ, he is a new creation: the old has gone, the new has come! (2 Corinthians 5:17).
The “selection process” is rigorous. Much is required of those who have been adopted into God’s family and who seek to serve Him, both now in this world, and later, in the world to come. As we declare the good news of God’s kingdom and point others to Christ Jesus, we belong to this kingdom of priests now, eagerly awaiting that time when we will appear with Christin glory.
If then you have been raised with Christ, seek the things that are above, where Christ is, seated at the right hand of God. Set your minds on things that are above, not on things that are on earth. For you have died, and your life is hidden with Christ in God. When Christ who is your life appears, then you also will appear with him in glory. (Colossians 3:1-4).
From the above I can only conclude that God intends for all who serve Him now, as ambassadors of the kingdom of God, to fulfil a priestly role here on earth by declaring the praises of Him who has called us out of darkness into His wonderful light.
In answer to your question, yes, through our active faith in Christ Jesus, we are part of this spiritual kingdom of priests. The glorious reality of this Kingdom of Priests, this Royal Priesthood, will become manifest when we go to be with the Lord.
Edit to clarify "The selection process is rigorous": I do not intend to suggest we have to work at our salvation in order to earn it. Salvation is not a process. It happens in a moment, entirely due to God's initiative. Rather than just delete that sentence I thought to explain myself better.