Fun fact: it doesn't actually take that much time without forwards acceleration for ullage to become an issue. For example, the Saturn V included solid-fuel rockets on the interstage between the first and second stages to provide enough forwards acceleration for the second stage engines to ignite, even though the gap between the first stage engines shutting down and the second stage engines igniting wasn't more than several seconds [0, page 14]. You can see the exhaust from those motors on the outside of this video of the Apollo 4 first stage separation [1].
Another fun fact: instead of using smaller motors to provide acceleration between stages, (some?) Soviet rockets ignited upper-stage engines while the lower-stage engines were still firing ("hot-staging"). This is why some soviet rockets had a lattice (or lattices) between stages instead of solid metal.
Yet another fun fact: it doesn't take much acceleration to sufficiently settle the fuel for ignition. The Saturn V's S-IVB stage used its equivalent of its RCS system, which is normally used for attitude control, to settle its fuel prior to firing for trans-lunar injection.
If you're interested in installing KSP mods, RealFuels includes ullage simulation, so you'll need to have some method of providing acceleration to get larger rocket motors to ignite reliably.
[0]: http://web.mit.edu/digitalapollo/Documents/Chapter5/saturnas... [1]: https://youtu.be/iJIY4n5oXjc?t=8
Edit: Fix description of hot-staging for Soviet rockets. Some Soviet rockets hot-staged their third stage and not their second stage, making the description incorrect.
http://www.pmdtechnology.com/Index.html
For very low accelerations, they can even draw propellant "upwards" against the acceleration.
https://forum.kerbalspaceprogram.com/index.php?/topic/168424...
I think a bigger issue might be that all the mass is at the engines, so the engines would stay nearly fixed with the other end of the rocket rotating rapidly, and pretty much all the propellant would go to the wrong end of the rocket.
https://en.wikipedia.org/wiki/Reaction_wheel
Basically, an heavy enclosed wheel that you use a motor to spin. Because of the conservation of angular momentum, the rocket will spin in the opposite direction (more slowly, because it's (probably) heavier).
I hadn't even considered that the fuel will just be floating in the tank with no forwards pressure
#ThingsKSPdoesn'tTeachYou