Single-walled carbon nanotubes (SWCNTs) are promising absorbers and emitters to enable novel photonic and optoelectronic applications but are also known to severely suffer from low optical quantum yields. Here we demonstrate SWCNTs excitons coupled to plasmonic nanocavities reaching deeply into the Purcell regime with FP=234 (average FP=76), near unity quantum yields of 70% (average 41%), and a photon emission rate of 1.7 MHz into the first lens. The measured ultra-narrow exciton linewidth (18 micro eV) implies furthermore generation of indistinguishable single photons from a SWCNT. To demonstrate utility beyond quantum light sources we show that nanocavity-coupled SWCNTs perform as single-molecule thermometers detecting plasmonically induced heat (Delta T=150K) in a unique interplay of excitons, phonons, and plasmons at the nanoscale.