Both players resolve their upkeep effects simultaneously, with each player deciding the order in which they want to resolve the things affecting them. Initiative doesn't matter at all during this phase except if you have a timing issue. A timing issue being if you need to decide when something affecting your stuff happens relative to something affecting your opponent's. We'll come back to this in a moment.
When you remove the last dissipate token, it triggers two things. One being the card's destruction, and the other being an attack made by the card. Both of these things occur "when" the dissipate token is removed. This means they are both trying to occur at the same time. Because both affect the same object, the Pillar of Righteous Flame (one granting it an attack and the other destroying it) the person who controls the Pillar gets to decide what order to resolve them in.
That's the trick to a lot of timing things in the Upkeep Phase. Identifying what is being affected by the thing, because whomever controls the object being affected is the one who gets to decide the timing. Wispwillow Amulet is another good example of this. When it removes it's last dissipate token, the Mage wearing it gains 1 Mana and the Amulet gets destroyed. One thing affects the Mage and the other affects the Amulet, but due to the unique rules regarding equipment both of these things will always have the same controller. Therefore, the same person gets to decide what order they occur in.
Lullaby is different. It has an effect that is affecting the creature it is attached to, while its destruction is obviously affecting the Lullaby card itself. Often Lullaby and the attached creature will have different controllers. This is a case where you're likely to have a timing issue as you want to decide which happens first. Does my Lullaby get destroyed first or does your creature get affected by it first? Because they are controlled by different people, this is a case of a timing isssue where the player who has Initiative for the round would decide.