Not under RAW
From the rules quoted in the question.
A spell that takes 1 round to cast is a full-round action. It comes into effect just before the beginning of your turn in the round after you began casting the spell. You then act normally after the spell is completed.
This seems pretty clear to me that a spell does not take effect in the same round as you cast it. Furthermore, actions (beyond the full round action to begin casting the spell) aren’t mentioned at all in the rule, so I’m not sure why an extra move action would influence things at all.
The sequence is:
- A combat round (Round N) begins
- You begin casting a spell on round N. (Either on your turn or out of turn with a spell or ability that lets you act out of turn)
- Round N ends, and Round N+1 begins
- When your turn comes up, the spell takes effect immediately before it.
Is there any interaction at all?
It’s unclear whether you even get to use the move action from Mythic Haste in conjunction with a full round action at all.
The rules for Full Round Actions state
A full-round action requires an entire round to complete. Thus, it can’t be coupled with a standard or a move action, though if it does not involve moving any distance, you can take a 5-foot step.
I think under a strict reading of this rule, taking a full round action prevents you from taking any move actions. You still have your move actions “unspent” (both your normal one and your bonus one from Mythic Haste) but are unable to use them and a Full Round Action in the same turn.
Under a looser reading, the Full Round Action prevents you from taking your normal move action, but has no effect on the “additional” move action granted by Mythic Haste.
I personally as GM would allow players to take the Move Action from Mythic Haste in conjunction with a Full Round Action, but I think the exact text of RAW favors the stricter interpretation.