An assembly is provided for an aircraft. This aircraft assembly includes a powerplant. The powerplant includes a gas turbine engine, an exhaust nozzle and a flowpath extending from the gas turbine engine to the exhaust nozzle. The exhaust nozzle is movable between a first position and a second position. The exhaust nozzle is configured to exhaust combustion products, received through the flowpath from the gas turbine engine, when the exhaust nozzle is in the first position. The exhaust nozzle is configured to block flow of the combustion products through the flowpath when the exhaust nozzle is in the second position.