I know quite well that the plethora of Protestant denominations means that there are almost no universal traditions. However, there are a number of particular traditions or beliefs that are held by the majority of Protestant denominations. In this sense, then, I'm wondering if there is a general consensus - and what the consensus is - as to who can baptize a new believer. An even better answer would be one that broadly categorizes the denominations that have a particular restriction on who can baptize and the denominations that don't have such a restriction. (Brownie points for an answer with a Wesleyan viewpoint.)
The Scriptures make no restriction on who can baptize another person. In the Great Commission of Matthew 28, Jesus tells His disciples to make disciples, to baptize them and to teach them obedience. It is very plausible to presume, then, that the one who disciples is the one who has the right to baptize and teach obedience as well. Yet, individuals can play different roles in this.
So, some Protestant denominations may restrict the privilege to pastors, but when they do so, it is likely not based on clear prohibitions of Scripture. Many Protestant denominations reject the idea of a clergy class and a laity class, so what a pastor can do, others who are not pastors can also do (except when it comes to legal issues like performing weddings).
Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, 20 teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age.” Matthew 28:19-20 NAS
In my church, which is a nondenominational Protestant church, a pastor or male member of the church will baptise a new believer. Often a child's father will baptise them. However, I don't know if this is a rule or just what usually happens