How can I rationalise the different basicities of cyclohexylamine, piperidine and morpholine?
You are right that (I) &lt; (II) and (II) > (III), but my explanation for (I) &lt; (II) is different from yours: nitrogen atom in (II) has more alkyl group than (I), therefore the its conjugated acid is more stable.

As inductive effect decrease fast along the carbon chain, even though cyclohexyl group have more carbons, its effect is still weaker than two alkyl group (you can treat piperidine as ethylpropylamine, they're similar). Inductive effect is usually stronger than hyperconjugation.

In (III), both two branches have -C effect, as oxygen is an electronegative element, which destabilises the conjugated acid. In (I), the only alkyl group has +C effect, which stabilise the conjugated acid. Therefore, (I) is a stronger base than (III)

Hence, we have the sequence of basicity: (II) > (I) > (III).

We can look up pKb of each substance to check the result:

Cyclohexylamine (I): 3.36
Piperidine (II): 2.78
Morpholine (III): 5.64

So, my comparison is rational.