It seems by Discovery Season 3 they're aware of the Kelvin timeline.
Consider the following from Season 3 Episode 9:
Kovich : Consider yourself lucky to have skipped the Temporal Wars. Among the many horrible things we discovered when weaponizing time: temporal travel can make you pretty sick. Turns out our molecules are designed to function in the time in which they're created.
Culber: But everyone on Discovery traveled through time.
Kovich: Yes, but only one of you is also from a parallel universe. Yor here traveled forward from 2379 and across from an alternate universe created by the temporal incursion of a Romulan mining ship. Before Georgiou, Yor was the only individual known to have travelled across both time and dimensions.
It doesn't seem they're aware that Spock made it to the Kelvin timeline.
If they were aware of Spock's crossover, Georgiou won't be the only one to have travelled across both time and dimensions, correct?
We don't know the details of the temporal wars, but I highly doubt Spock's body would have remained in the Kelvin timeline. Even with temporal accords, couldn't they make an exception to return Spock's body to the Prime universe?