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Nisan comments on Problems of the Deutsch-Wallace version of Many Worlds - Less Wrong Discussion
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Ah, I thought of a charitable interpretation of "there is a unique (up to affine transformations) utility function over the rewards". Given a preference ordering on sequences of rewards, there is a unique utility function on individual rewards that recovers that preference ordering. I believe this because if rewards are repeatable, the diachronicity hypothesis implies that any utility function on sequences of rewards must be additive. (We also need a hypothesis ruling out lexicographically-ordered preferences.)
You're right. Diachronic consistency is required to establish the uniqueness of the utility function. Also, Wallace does include continuity axioms that rule out lexically ordered preferences, but I left them out of my summary for the sake of simplicity.