If a solar project including new roof started in December 2023 and finished in Jan 2024 be claimed in both tax year's returns? Or just one? Sales rep indicated both.

You can only claim the credit in the year the solar installation was completed and paid for.  That would be 2024.

Note - The roof replacement is not part of the solar installation.  You add the cost of the roof replacement to the basis of the home when it is sold.

What if the system was installed but not turned on until January because need for roof replacement delaying submission of system to power company for activation?  Is that still just 2024?

Still, only in 2024 since that is when it became operational.