So I have an seen the equation to calculate precession but it breaks down at when there is not sufficient angular momentum, for example if an object is not spinning we divide by zero and the precession speed becomes infinite.
Is there are better/separate equation that allows us to tell if precession will occur and how much it will be impacted by the downwards torque?
The expression for rate of precession that you refer to is an approximation, the validity of that approximation is limited to cases where the spin rate is high enough such that any resulting gyroscopic precession is far slower than the rate of spin.
There is a 2012 answer by me where I discuss the mechanism of gyroscopic precesssion. The approach to explanation in that discussion does not use the concept of angular momentum; instead the explanation capitalizes on symmetry.
As we know: for a given amount of torque exerted on the spinning object: the faster the rate of spin the slower the corresponding gyroscopic precession.
For the approximation the relevant factor is the ratio of amount of torque and the rate of spin. If the torque is small relative to the rate of spin then for practical purposes the approximation suffices.
To figure what happens when the torque is not small (relative to the rate or spin), think of how a spinning top responds to applied torque.
We have that when the spinning top is spinning fast, and torque is introduced, the response of the spinning top is to proceed to gyroscopic precession. Importantly: while usually not visible to the naked eye, the spinning top does yield a little to the torque.
In classroom demonstrations the spin rate is always made very high, and because of that it looks as if the gyro wheel starts precessing instead of yielding to the torque. However, some yield must occur; to not yield to the torque at all would be a violation of the laws of motion.
This aspect of yielding to the torque a little is the subject of a tabletop experiment by Svilen Kostov and Daniel Hammer. The titel of their article is: It has to go down a little, in order to go around. I strongly recommend studying the result they obtained
As you try slower and slower spin rates:
Keep the magnitude of the torque that you add the same, and try ever slower spin rates. The slower the spin rate of the gyro wheel, the more the gyro wheel will yield to an applied torque. Pretty soon there isn't even time to reach any precession-like motion; the gyro wheel will flop over wildly when the torque is applied.
I want to emphasize: It's not that there is some cut-off spin rate, and that above it precession does occur and below it it doesn't.
If you create perfectly frictionless circumstances, and you make the torque small enough such that the resulting precession is slow compared to the spin rate then you would be able to go down to very, very slow spin rates and still get gyroscopic precession; it's about the ratio of applied torque to spin rate.
Conversely, as you make the magnitude of the torque higher and higher relative to the spin rate the response of the wheel looks less and less like gyroscopic precession.
Linear mechanics and angular mechanics
There are discussions of gyroscopic precession in circulation that suggest there is a 1-on-1 correspondence between mathematical expressions for linear mechanics and for angular mechanics. (With angular momentum for linear momentum and so on.) The expression for response of a gyroscope wheel to applied torque is then presented as a direct analogon of the corresponding expression for linear mechanics, and it is suggested that the expression is exactly valid.
As you point out: the equation breaks down when the amount of torque is outside of the range where the approximation is sufficient.
That observation refutes the assumption that all of the expressions for angular mechanics are in 1-on-1 correspondence with counterparts in linear mechanics.
