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Here's the specific sound I'm talking about (it starts specifically 56 seconds into the video, as well as at 2:02 for the second engine):

I've only really heard that sound on low-bypass engines (F/A-18s, BAE Hawks, etc); they don't occur right at engine-start (so as far as I'm aware it's not just bleed air being blown into a turbine for spin-up); I was thinking that it's perhaps either fuel flow causing an increase in exhaust flow through a slowly-moving turbine, or that it might be the hydraulic pump (or something similar) connected to the engine accessory drive.

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    $\begingroup$ You can hear that on high bypass engines, too. I believe this is related to flame stabilization in the burn chamber at low speeds. Combustion is not stationary causing pressure fluctuations and thus the sound $\endgroup$ – Jan Nov 11 '20 at 10:17
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Almost all jet engines will have one or more "resonance" conditions where the combination of turbine RPM, mechanical elasticity and aeroelasticity constructively reinforces to cause mechanical or aerodynamic vibration.

Think of it this way: If you have a slinky spring (mechanically elastic) toy you can hold it by one end and let the rest drop freely below that. Now if you move your hand up and down at different frequencies (up and down and back up again in 5 seconds, 4 seconds, 3 seconds, 2 seconds, 1 second, 1/2 second) you will find that there is one rate that will cause the most deflection/movement of the spring toy even though you are moving your hand through exactly the same distance. This is the one closest to its resonance.

Now the ductwork and spools of a gas turbine engine are just a little more complicated than our Slinky toy, but still a mechanical (in this case, aero-mechanical) system susceptible to resonance conditions. The trick is to design the system so that these resonance conditions, which can be mechanically destructive, don't happen often or for very long. As the turbine RPM increases through the resonance range things start to vibrate (in this case at ~250-275 Hz) wildly enough to create that extra sound.

In fact, the designers of that GE F414 engine could tell you exactly the RPM range where that happens. The aviators and groundcrew are attune to the sounds of this machine and would likely shut things down and have a closer look if this particular phenomenon didn't happen as expected.

So, the horn-like noise is vibration of the mechanical parts of the engine ductwork and the air flowing through it for the short period of time that the engine RPM and airflow align to make this condition happen.

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