It depends who you ask.
If you ask the Catholics, the 2010 catholic.com article Was James the Real Leader of the Early Church? makes the case that despite Acts 15 appearance that James was the leader, if we take into account the number of times Peter's name appear in the whole book (54) compared to James (4) and the prominence of Peter in the Gospels, Acts 15 alone wouldn't be sufficient to establish James as the leader.
The 3 theories of authorities mentioned in the article are:
1) Peter and the other members of the twelve were concerned with a Christian mission far more extensive than just Jerusalem. They were never really local church leaders, once Jerusalem became big enough to require such caretakers. James was the first leader of the local church at Jerusalem (at least for the Hebrew Christians) and remained there after Peter and the other members of the twelve left the scene, whether through death or on travels. James had authority only in Jerusalem (and its “province”), but his name was known more widely because he was a blood relative of Jesus. Paul’s loyalty was to the “mother church” or community of saints in Jerusalem. His respect for James was a respect for the local leader of that church.
2) Peter was a local leader at Jerusalem (even though he was known more widely because he had been a close follower of Jesus during the ministry). James took Peter’s place as the local Jerusalem leader (when Peter left Jerusalem or even earlier). Neither of them had a role as leader in the Universal Church, for, in fact, there was no single leader in the Universal Church.
3) Peter was a universal leader, operating from Jerusalem as the center of Christianity, and was succeeded by James. In other words, the position of universal influence that Peter had at Jerusalem (except his apostleship) was transferred to James when Peter left Jerusalem or even earlier.
The first theory aligns essentially with the Catholic belief; the second covers a wide range of mainline Protestant perspectives; and the third—the most extreme view—is embraced by more radical, liberal scholars.
The article noted of the "unlikely allies" supporting theory #2: Fundamentalist and Liberal scholars, both have strong anti-authoritarian sentiments and thus rallied against the Catholic theory #1.
At any rate, the article tries to be even handed by referencing scholarly books supporting each theory so the readers can decide for themselves.