Every explanation of diodes I can find say current cannot meaningfully flow from the cathode to the anode.
There are at least two reverse current flows, but most diode users are able to ignore them (until the application demands that they can't)
However an NPN transistor is fundamentally two diodes and the only way it can work is if one of the diodes allows current to flow cathode to anode.
A BJT (bipolar junction transistor) is not two diodes. It contains two closely spaced junctions, which are similar to the single junctions you find in diodes. If you bias these junctions one at a time, then they behave like diodes. The junctions are so closely spaced however that the 3-terminal behaviour is much more complicated than the '2 diodes connected together' picture that's drawn as a 'lie to children'. If you want to push the model, then you could argue that the operation of the BE diode modifies the operation of the CB diode, because of the way charge in one can move into the other, but that's a dead-end really as it poses more questions than it answers. A BJT is a three terminal device, understand it through all three.
When a diode is reverse biassed, a reverse current can flow, usually written as IS. This is usually very small, pA or fA levels are typical for small signal diodes. That's not a meaningful current for many people or applications, but if you're doing doing a design with those current levels, you need to design for it. This current is not a defect, it appears in the Shockley diode equation which describes the ideal diode current flow at all bias voltages.
When a junction diode is forward biassed, the junction fills with carriers. When the diode is then suddenly reverse biassed, these carriers have to be swept out of the junction, and this movement of carriers is a current, which can be very large. This is the reverse recovery current, and will usually last us to ns before the charge is depleted. The current will then suddenly cease, or more accurately, drop to the IS level, often causing a very sharp voltage transient with signal components into the MHz or GHz. The 1N400x rectifier diode series are notorius for having a very large charge storage, and could often cause noise when used as the rectifier for audio amplifiers.
In your particular diagram, both of these diode conduction mechanisms would appear to be irrelevant, with the motor implying 'large' currents, and 'long' time constants.