There is an order given for the Brahma Viharas. But I don't think I have read that the order was specifically given as to how to develop them. If it was given in an order how to develop them, it is in the wrong order.
The Pali words are Metta, Karuna, Mudita and Upekkha. Now, going backwards, Upekkha is Equanimity, Mudita is Sympathetic Joy, Karuna is Compassion and Metta is normally called Lovingkindness, however we define Metta as Compassion and Lovingkindness. So, there are the four.
Which one of the four would you think could be termed the first one to develop? The easiest to develop maybe we could say, the best one to develop we could say, the one most important to develop we could say? Now since the first one in the order is Metta (Compassion and Lovingkindness), but that doesn't make any sense because Compassion is listed after it. Metta and Karuna need to swap, so we develop Compassion first, then we develop Compassion and Lovingkindness after Compassion. In this way, if the order was given as to how to develop them, it would make sense to switch those two. Sympathetic Joy is extremely hard to develop, so to me it comes after Compassion and Lovingkindness. Equanimity is the trickiest to develop, so let's keep it at the end.