The thrust acts on the nozzle and combustion chamber walls by virtue of the pressure differential they contain. Using the RS-25 (SSME) as an example, with illustrations from this pdf, I can highlight a few of the components.
Page 11 (pdf page 17) shows the gimbal bearing assembly, a ball-and-socket joint assembly made from a titanium forging. That is where more or less all of the thrust load is transferred to the vehicles thrust structure.
Page 23 (pdf page 29) shows the Block IIA powerhead, which is the combustion chamber and turbopump assembly. At the top center of the combustion chamber, you can see a flower-shaped flange that is bolted to the gimbal bearing assembly and is where load is transferred from the combustion chamber/nozzle assembly. You can also see this on Page 26 (pdf page 32) and in closer detail on Page 42 (pdf page 48) -- note the term "Thrust Cone".
Pages 45 and 46 (pdf pages 51 and 52) show detail of the combustion chamber. Load from the nozzle is sent up mostly through the throat ring, except not really, because it's also being reacted by the actual exhaust gas present in the combustion chamber. In all likelihood, the throat ring is probably under tension when the engine is firing.
The nozzle detail is seen on Pages 49 and 51 (pdf pages 55 and 57), you can see the flange through which it attaches to the combustion chamber at the top of the illustration. The Hatbands react the hoop stresses generated by the pressure inside the nozzle and prevent it from just blowing apart, much like the iron bands on a whiskey barrel. The conical(ish) shape of the nozzle means that the net result of the pressure is directed upward.