Funnily enough, we've never seen any Trek universe feature noticeably changed when it appears the second time.
Kirk's ship is exactly the same no matter how often she reappears. When Kirk and Spock and McCoy are recast, it's with maximum visual commonality in mind (although eye color is too much of a hassle, and voices may not be perfect). An artifact from another Trek era looks exactly like other artifacts from that era, regardless of whether it's an actual historical prop unearthed from some basement archive, or an all-new construct.
I wouldn't expect the Botany Bay to look any different from how she looked in TOS, then. Except in the sense that TOS had her at "low resolution", and TOS-R gave "more resolution", so a bit more detail and texture but zero change in shape or the like.
Then again, ST:ID does feature the one and only Trek departure from original material committed so far: Khan's crew are in separate sarcophagi, when in TOS they were resting on shelves. Transfer ought not to have been medically possible... And why would Khan have 72 coffins lying around if he already had shelves?
(But perhaps Khan always was on a shelf, and all the 72 surviving crew were in sarcophagi, while everybody else seen on a shelf in TOS was dead?)
Timo Saloiniemi
