If there was an offer and an acceptance, and if there was consideration given...I don't know the details of the arrangement...then yes there was a bailment but almost certainly for an indefinite period. The owner of the logbook gave notice he wanted it returned and any bailment that previously existed was therefore terminated at the time the notice was given.
For conversion to be made out, the flying school would have to deal in the logbook for its own benefit without authority from what I remember from first year law school. If they deal in it, then they have appropriated the rights of the owner but proving this was dishonestly done is difficult and turns very much on the particular facts.
If they haven't dealt with the logbook, then trespass to goods seems a better way to go because it is simply an action for the return of goods withheld from the true owner.
But this is a civil action that has to be taken by the owner of the logbook. Anyone contemplating a civil action has to take into account the potential costs associated with losing it. Even if the owner self-represents they could still be up for an adverse award of costs in that event. They would need to look very carefully at any and all contracts entered into with the flying school to ensure there was no other contractual arrangement governing rights wrt the logbook.
40 years ago when I started lessons my instructor filled out my logbook in meticulous fashion entering the section numbers for each of the skills completed. But I kept it at home and still have it. No arguments then and instruction professionally given by a reputable and honest organisation.
All of the foregoing is a discussion without retainer and is purely my opinion about a scenario the details of which I have only scant knowledge. It does not constitute legal advice and should not be construed as such. If you need legal advice you should instruct a solicitor privately.
Kaz