Christ is the head
The clearest starting point is Jesus. He promised to build his church, and the New Testament says God made him head over the church, which is described as his body (Matthew 16:18; Ephesians 1:22-23). That means no earthly congregation, city, bishop, or tradition replaces Christ as the church's ultimate foundation and leader.
The word catholic, with a small c, means universal. In that broad historical sense, Christians sometimes speak of the catholic church as all who belong to Christ across places and generations. But the Bible's emphasis is not on one later institution being called the mother of every other church. It is on believers being joined to Christ and to one another.
And hath put all things under his feet, and gave him to be the head over all things to the church, Which is his body, the fulness of him that filleth all in all.
Jerusalem had an early leading role
If by mother church someone means the first major centre of the earliest Christian movement, Jerusalem is the strongest biblical answer. The apostles began there after Jesus' resurrection, and believers there shared life, teaching, prayer, and practical care (Acts 2:42-47).
Later, when Gentile Christians and Jewish Christians faced a serious disagreement about circumcision, Paul and Barnabas went to Jerusalem to discuss it with the apostles and elders (Acts 15:1-2). The Jerusalem leaders helped the churches seek a faithful, united response. Yet the decision was communicated to several churches, not presented as one city ruling Christ's whole body (Acts 15:22-23).
And certain men which came down from Judaea taught the brethren, and said, Except ye be circumcised after the manner of Moses, ye cannot be saved. When therefore Paul and Barnabas had no small dissension and disputation with them, they determined that Paul and Barnabas, and certain other of them, should go up to Jerusalem unto the apostles and elders about this question.
What about Rome?
The church in Rome was clearly significant. Paul wrote a substantial letter to its Christians, and he thanked God for their widely known faith (Romans 1:7-8). Peter and Paul are both closely associated with Rome in early Christian history, and the Roman Catholic Church sees continuity with the apostles as deeply important.
Still, the New Testament does not explicitly name the church in Rome as the mother church. Paul's letter addresses a community of saints, and he speaks of his desire to visit and strengthen them, which shows mutual ministry among believers rather than using the title mother church for Rome (Romans 1:11-12).
Christians see church authority differently
Catholic Christians understand the Bishop of Rome, the Pope, to have a special pastoral role in the universal church, connected with Peter's place among the apostles. Other Christians honour the ancient church of Rome and value its history, but do not believe Scripture gives later Roman bishops authority over every Christian congregation. This is a real disagreement, and it should be discussed honestly and graciously.
Whatever tradition a Christian belongs to, the New Testament calls believers to humility, love, and a serious effort to preserve unity in Christ (Ephesians 4:3-6). The question is not only which church has the oldest claims. It is also whether we are listening to Christ, trusting his gospel, and loving his people.
Endeavouring to keep the unity of the Spirit in the bond of peace. There is one body, and one Spirit, even as ye are called in one hope of your calling; One Lord, one faith, one baptism, One God and Father of all, who is above all, and through all, and in you all.