Compare:
(1) "During the Upkeep Phase, while Skeelax is in the same zone as an object with a Burn Condition, he gains the Regenerate 2 trait."
(2) "During the Upkeep Phase, if Skeelax is in the same zone as an object with a Burn Condition, he gains the Regenerate 2 trait until the end of the Phase."
Since the actual wording (1) is a "while" condition, it only applies as long as there is an object "burning." As soon as it stops burning (e.g. the burning object is destroyed or the burn rolls a blank in absence of [mwcard=MWSTX2FFE01]Adramelech's Touch[/mwcard]), he no longer has Regenerate 2. On the other hand, if the wording were an "if" condition (as in (2)), then one could argue that it applies as long as an object was burning at some point during the Upkeep Phase, even after any existing burn tokens are removed, regardless of who has initiative. That is, you check to see if the condition is satisfied; if it is, he has Regenerate, regardless of whether or not the condition changes during the Phase.
(The extra bit I appended to the end of (2) just makes sure his Regenerate 2 goes away if Skeelax leaves the zone or there are no burn tokens in his zone in future game rounds; otherwise it might stay "on" for the rest of the game!)
Then again, it depends on
when you check this condition. If you do not have initiative, you have to wait for the other player to upkeep his effects and
then check if your condition is met. That is not different from you first sentence's wording.
If we look at the big picture here, I don't see how this situation is different from any of the following:
- Creature with regenerate and a burn token
- Creature enchanted with Ghoul Rot and has regenerate
- Cursed Creature with regenerate and Necromancer uses his ability to pay one mana and deal extra damage (forgot the ability's name)
- Cursed creature with regenerate and Warlock uses [mwcard=MW1Q21]Moloch's Torment[/mwcard] to deal one damage.
-Druid using his Treebond Ability and regenerating and taking direct poison damage from Ghoul and from [mwcard=MW1Q21]Moloch's Torment[/mwcard]
- Mage wearing a [mwcard=MWSTX1CKQ01]Sunfire Amulet[/mwcard] and receiving any kind of direct damage during the upkeep.
If I'm not mistaken, all these situations are dealt with initiative: for instance, if a player without initative loses his creature by direct damage, it just can't regenerate.
It seems to me that, for consistency issues, we have to play it that way because this can resolve a lot of issues that can arise as new spells keep coming in.
Thematically, it also makes sens that initiative rules effects during the upkeep as it is the controlling mage who decides when to burn or poison damage a creature.