Hunting out more references... the Tale of Years in HoME XII tells of Elrond's children:
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Originally Posted by TY 2300
These children were three parts of Elven-race, but the doom spoken at their birth was that they should live even as the Elves so long as their father remained in Middle-earth; but if he departed they should have then the choice either to pass over the Sea with him, or to become mortal, if they remained behind.
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This contradicts the various NoME texts which claim Arwen's choice was made at her wedding, and also implies the El-twins both became mortal.
The beginning of HoME XII's "The Making of Appendix A" holds that Elros' choice immediately bound his descendents, while to Elrond's children...
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Originally Posted by Making of Appendix A
... a choice was also appointed: to pass with him from the circles of the world; or if they wedded with one of Mankind, to become mortal and die in Middle-earth.
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It's unclear what would happen if they stayed in Middle-earth unmarried in this version.
I vaguely recognise the reference
William Cloud Hicklin mentions, but also can't find the cite.
EDIT: Thought it might be in Letters, but no. Letters 153 & 154 both assert that "the half-elven" were given a choice, and name both Elrond and Arwen. If linked with WCH's cite, that would imply that all children of an elf and a half-elf would have the same choice, while all children of a mortal and a half-elf would be mortal. But it might also just be a summary for simplicity.
hS