According to Islamic teachings, Rukhsati, which refers to the bride leaving her family home and moving in with her husband, is not a requirement immediately after the Nikah, or marriage contract, is performed. This is evident from the marriage of Hazrat Ali and Hazrat Fatimah, where Rukhsati took place a year after the Nikah.
Regarding the situation you have described, if both parties agree to delay Rukhsati until after Ramadan, then it is permissible as there is no Islamic injunction that requires immediate Rukhsati after Nikah.
As for sexual relations, Islam permits it after the Nikah is performed. However, it is not considered socially acceptable to engage in sexual relations before Rukhsati. Therefore, it is recommended to wait until after Rukhsati to consummate the marriage.