Surely this manuscript on gunpowder-based fireworks rockets
by Conrad Haas (1509–1576) qualifies?
Or at the very least the followup work on multistage rockets by Johann Schmidlap who published "Künstliche und rechtschaffene Fewrwerck zum Schimpff,” firstly printed in Nuremberg in 1561?
That diagram clearly shows a divergent nozzle on a gunpowder rocket....
Probably units and the format of eq (7) are the problem. After looking at the referenced website by Nakka, I used the questions information given to get $\dot m= 1.187 kg/s$ (using a rounded off r=12mm/s= .012m/s). Also calculated throat pressure and temperature of 3421000 Pa and 1616 K. Equation (7) should have parentheses around the $ w_t/P_t$,because the ...
You are omitting an important part of the force exerted by a rocket engine. A better expression is
$$F = \dot m v_e + (P_e-P_a)A_e$$
In this expression,
$F$ is the force exerted by the engine on the rocket body,
$\dot m$ is the mass flow rate of exhaust exiting the engine,
$v_e$ is the velocity of the exhaust gas at the exit plane,
$P_e$ is the pressure of ...