A hybrid nanostructure where a graphene nanoflake (GNF) is optically coupled to a carbon nanotube (CNT) could potentially possess enhanced sensing capabilities compared to the individual constituents whilst inheriting their high biocompatibility, favourable electrical, mechanical and spectroscopic properties. Therefore, in this paper, we investigate the scattering characteristics of an all-carbon exciton-plasmon nanohybrid which was made by coupling an elliptical GNF resonator to a semiconducting CNT gain element. We analytically model the nanohybrid as an open quantum system using cavity quantum electrodynamics. We derive analytical expressions for the dipole moment operator and the dipole response field of the GNF and characterize the Rayleigh scattering spectrum of the nanohybrid. These analytical expressions are valid for any arbitrary ellipsoidal nanoresonator coupled to a quantum emitter. Furthermore, we perform a detailed numerical analysis, the results of which indicate that the GNF-CNT nanohybrid exhibits enhanced and versatile scattering capabilities compared to the individual constituents. We show that the spectral signatures of the nanohybrid are highly tunable using a multitude of system parameters such as Fermi energy of the GNF, component dimensions, GNF-CNT separation distance and the permittivity of the submerging medium. We finally demonstrate the prospect of using the proposed nanohybrid to reconstruct the permittivity profile of a tumour. The high biocompatibility and high sensitivity to the dielectric properties of the environment make the proposed GNF-CNT nanohybrid an ideal candidate for such biosensing applications.